;-NRLF 


I 


1 


UNIVERSITY  OF  CALIFORNIA 

PEFARTMENT  OF  CIVIL  ENGINEERING 

BERKELEY,  CALIFORNIA 


Engineering 
Library 


A   HISTORY   OF 
ELEMENTARY   MATHEMATICS 


THE  MACMILLAN  COMPANY 

NEW  YORK    •    BOSTON    •    CHICAGO    •    DALLAS 
ATLANTA   •    SAN    FRANCISCO 

MACMILLAN  &  CO.,  LIMITED 

LONDON   •    BOMBAY   •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


A   HISTORY 


OF 


ELEMENTARY   MATHEMATICS 

WITH 

HINTS   ON   METHODS   OF   TEACHING 


BY 


FLORIAN    CAJORI,   PH.D. 

PROFESSOR    OF    MATHEMATICS    IN    COLORADO    COLLEGE 


REVISED  AND  ENLARGED  EDITION 


THE   MACMILLAN   COMPANY 

LONDON  :   MACMILLAN  &  CO.,  LTD. 
1917 

All  rights  reserved 


Engineering 
Library 


COPYRIGHT,  1896  AND  1917, 
BY  THE  MACMILLAN  COMPANY. 


Set  up  and  electrotyped  September,  1896.     Reprinted  August,  1897; 
March,  1905;  October,  1907;  August,  1910;   February,  1914. 
Revised  and  enlarged  edition,  February,  1917. 


•o   ^ 


PREFACE   TO   THE   FIRST   EDITION 


"THE  education  of  the  child  must  accord  both  in  mode 
and  arrangement  with  the  education  of  mankind  as  consid- 
ered historically ;  or,  in  other  words,  the  genesis  of  knowledge 
in  the  individual  must  follow  the  same  course  as  the  genesis 
of  knowledge  in  the  race.  To  M.  Comte  we  believe  society 
owes  the  enunciation  of  this  doctrine  —  a  doctrine  which  we 
may  accept  without  committing  ourselves  to  his  theory  of 
the  genesis  of  knowledge,  either  in  its  causes  or  its  order."1 
If  this  principle,  held  also  by  Pestalozzi  and  Froebel,  be 
correct,  then  it  would  seem  as  if  the  knowledge  of  the 
history  of  a  science  must  be  an  effectual  aid  in  teaching 
that  science.  Be  this  doctrine  true  or  false,  certainly  the 
experience  of  many  instructors  establishes  the  importance 
of  mathematical  history  in  teaching.2  With  the  hope  of 
being  of  some  assistance  to  my  fellow-teachers,  I  have  pre- 
pared this  book  and  have  interlined  my  narrative  with 
occasional  remarks  and  suggestions  on  methods  of  teaching. 
No  doubt,  the  thoughtful  reader  will  draw  many  useful 

1  HERBERT  SPENCER,  Education :  Intellectual,  Moral,  and  Physical 
New  York,  1894,  p.  122.     See  also  R.  H.  QUICK,  Educational  Reformers, 
1879,  p.  191. 

2  See  G.  HEPPEL,  "The  Use  of  History  in  Teaching  Mathematics," 
Nature,  Vol.  48,  1893,  pp.  16-18. 

v 


OOOOOO 


Vi  PREFACE 

lessons  from   the  study  of  mathematical   history   which  are 
not  directly  pointed  out  in  the  text. 

In  the  preparation  of  this  history,  I  have  made  extensive 
use  of  the  works  of  Cantor,  Hankel,  linger,  De  Morgan,  Pea- 
cock, Gow,  Allman,  Loria,  and  of  other  prominent  writers 
on  the  history  of  mathematics.  Original  sources  have  been 
consulted,  whenever  opportunity  has  presented  itself.  It 
gives  me  much  pleasure  to  acknowledge  the  assistance  ren- 
dered by  the  United  States  Bureau  of  Education,  in  for- 
warding for  examination  a  number  of  old  text-books  which 
otherwise  would  have  been  inaccessible  to  me.  It  should 
also  be  said  that  a  large  number  of  passages  in  this  book 
are  taken,  with  only  slight  alteration,  from  my  History  of 
Mathematics,  Macmillan  &  Co.,  1895.  .  .  . 

FLORIAN  CAJORI. 

COLORADO  COLLEGE,  COLORADO  SPRINGS, 
July,  1896. 

PREFACE   TO   THE   SECOND  EDITION 

IN  the  endeavor  to  bring  this  history  down  to  date,  numer- 
ous alterations  and  additions  have  been  made. 

FLORIAN  CAJORI. 
COLORADO  COLLEGE, 
December,  1916. 


CONTENTS 


PAQ« 

ANTIQUITY 1 

NUMBER-SYSTEMS  AND  NUMERALS 1 

ARITHMETIC  AND  ALGEBRA 19 

Egypt       .        .      ' 19 

Greece 26 

Home 37 

GEOMETRY  AND  TRIGONOMETRY 43 

Egypt  and  Babylonia      .        . 43 

Greece 46 

Borne 89 

MIDDLE   AGES 93 

ARITHMETIC  AND  ALGEBRA 93 

Hindus     .                 93 

Arabs                       : 103 

Europe  during  the  Middle  Ages Ill 

Introduction  of  Roman  Arithmetic Ill 

Translation  of  Arabic  Manuscripts 118 

The  First  Awakening 119 

GEOMETRY  AND  TRIGONOMETRY    .                  122 

Hindus    .        .        / 122 

Arabs       . % 125 

Europe  during  the  Middle  Ages     .        .        .        .        .        .  131 

Introduction  of  Roman  Geometry     .....  131 

Translation  of  Arabic  Manuscripts  .         .         .        .        .  132 

The  First  Awakening 134 

vii 


Vlll  CONTENTS 

PAGE 

MODERN  TIMES 139 

ARITHMETIC  .        *        .        .        .        .        .        .        .        .        .  139 

Its  Development  as  a  Science  and  Art  .....  139 

English  Weights  and  Measures       ......  167 

Rise  of  the  Commercial  School  in  England .        .        .         .  179 

Causes  which  Checked  the  Growth  of  Demonstrative  Arith- 
metic in  England .        .  204 

Reforms  in  Arithmetical  Teaching   >     .        .        .        .        .  211 

Arithmetic  in  the  United  States 215 

"Pleasant  and  Diverting  Questions" 219 

ALGEBRA 224 

The  Renaissance 224 

The  Last  Three  Centuries 234 

GEOMETRY  AND  TRIGONOMETRY 245 

Editions  of  Euclid.     Early  Researches .....  245 

The  Beginning  of  Modern  Synthetic  Geometry     .        .        .  252 

Modern  Elementary  Geometry 256 

Modern  Synthetic  Geometry 257 

Modern  Geometry  of  the  Triangle  and  Circle          .        .  259 

Non-Euclidean  Geometry 266 

Text-books  on  Elementary  Geometry        ....  275 

RECENT  MOVEMENTS  IN  TEACHING 290 

Tfie  Perry  Movement 291 

International  Commission 297 

American  Associations    ........  301 

Attacks  upon   the   Study  of  Mathematics  as  a   Training  of 

the  Mind    .  304 


A  HISTORY  OF  MATHEMATICS 


ANTIQUITY 


NUMBER-SYSTEMS   AND   NUMERALS 

NEARLY  all  number-systems,  both  ancient  and  modern,  are 
based  on  the  scale  of  5,  10,  or  20.  The  reason  for  this  it  is 
not  difficult  to  see.  When  a  child  learns  to  count,  he  makes 
use  of  his  fingers  and  perhaps  of  his  toes.  In  the  same  way 
the  savages  of  prehistoric  times  unquestionably  counted  on 
their  fingers  and  in  some  cases  also  on  their  toes.  Such  is 
indeed  the  practice  of  the  African,  the  Eskimo,  and  the  South 
Sea  Islander  of  to-day.1  This  recourse  to  the  fingers  has 
often  resulted  in  the  development  of  a  more  or  less  extended 
pantomime  number-system,  in  which  the  fingers  were  used  as 
in.  a  deaf  and  dumb  alphabet.1  Evidence  of  the  prevalence  of 
finger  symbolisms  is  found  among  the  ancient  Egyptians, 
Babylonians,  Greeks,  and  Romans,  as  also  among  the  Euro- 
peans of  the  middle  ages :  even  now  nearly  all  Eastern  nations 
use  finger  symbolisms.  The  Chinese  express  on  the  left  hand 

1  L.  L.  CONANT,  "Primitive  Number-Systems,"  in  Smithsonian  lie- 
port,  1892,  p.  584. 


*,«         .  * 

2    £'*^'  A   fr!ST?ORY-OF   MATHEMATICS 

"  all  numbers  less  than  100,000 ;  the  thumb  nail  of  the  right 
hand  touches  each  joint  of  the  little  finger,  passing  first  up 
the  external  side,  then  down  the  middle,  and  afterwards  up 
the  other  side  of  it,  in  order  to  express  the  nine  digits ;  the 
tens  are  denoted  in  the  same  way,  on  the  second  finger;  the 
hundreds  on  the  third ;  the  thousands  on  the  fourth ;  and  ten- 
thousands  on  the  thumb.  It  would  be  merely  necessary  to 
proceed  to  the  right  hand  in  order  to  be  able  to  extend  this 
system  of  numeration." l  So  common  is  the  use  of  this  finger- 
symbolism  that  traders  are  said  to  communicate  to  one  another 
the  price  at  which  they  are  willing  to  buy  or  sell  by  touching 
hands,  the  act  being  concealed  by  their  cloaks  from  observa- 
tion of  by-standers. 

Had  the  number  of  fingers  and  toes  been  different  in  man, 
then  the  prevalent  number-systems  of  the  world  would  have 
been  different  also.  We  are  safe  in  saying  that  had  one  more 
finger  sprouted  from  each  human  hand,  making  twelve  fingers 
in  all,  then  the  numerical  scale  adopted  by  civilized  nations 
would  not  be  the  decimal,  but  the  duodecimal.  Two  more 
symbols  would  be  necessary  to  represent  10  and  11,  respec- 
tively. As  far  as  arithmetic  is  concerned,  it  is  certainly  to  be 
regretted  that  a  sixth  finger  did  not  appear.  Except  for  the 
necessity  of  using  two  more  signs  or  numerals  and  of  being 
obliged  to  learn  the  multiplication  table  as  far  as  12  x  12,  the 
duodecimal  system  is  decidedly  superior  to  the  decimal.  The 
number  twelve  has  for  its  exact  divisors  2,  3,  4,  6,  while  ten  has 
only  2  and  5.  In  ordinary  business  affairs,  the  fractions  -J-,  £, •  J, 
are  used  extensively,  and  it  is  very  convenient  to  have  a  base 
which  is  an  exact  multiple  of  2,  3,  and  4.  Among  the  most 
zealous  advocates  of  the  duodecimal  scale  was  Charles  XII. 

1  GEORGE  PEACOCK,  article  "Arithmetic,"  in  Encyclopaedia  Metropoli- 
tana  (The  Encyclopedia  of  Pure  Mathematics},  p.  394.  Hereafter  we 
shall  cite  this  very  valuable  article  as  PEACOCK. 


NUMBER-SYSTEMS   AND   NUMERALS  3 

of  Sweden,  who,  at  the  time  of  his  death,  was  contemplating 
the  change  for  his  dominions  from  the  decimal  to  the  duo- 
decimal. 1  But  it  is  not  likely  that  the  change  will  ever  be 
brought  about.  So  deeply  rooted  is  the  decimal  system  that 
when  the  storm  of  the  French  Devolution  swept  out  of  exist- 
ence other  old  institutions,  the  decimal  system  not  only 
remained  unshaken,  but  was  more  firmly  established  than 
ever.  The  advantages  of  twelve  as  a  base  were  not  recognized 
until  arithmetic  was  so  far  developed  as  to  make  a  change 
impossible.  "The  case  is  the  not  uncommon  one  of  high 
civilization  bearing  evident  traces  of  the  rudeness  of  its  origin 
in  ancient  barbaric  life." 2 

Of  the  notations  based  on  human  anatomy,  the  quinary  and 
vigesimal  systems  are  frequent  among  the  lower  races,  while 
the  higher  nations  have  usually  avoided  the  one  as  too  scanty 
and  the  other  as  too  cumbrous,  preferring  the  intermediate 
decimal  system.3  Peoples  have  not  always  consistently 
adhered  to  any  one  scale.  In  the  quinary  system,  5,  25,  125, 
625,  etc.,  should  be  the  values  of  the  successive  higher  units, 
but  a  quinary  system  thus  carried  out  was  never  in  actual  use : 
whenever  it  was  extended  to  higher  numbers  it  invariably  ran 
either  into  the  decimal  or  into  the  vigesimal  system.  "  The 
home  par  excellence  of  the  quinary,  or  rather  of  the  quinary- 
vigesimal  scale,  is  America.  It  is  practically  universal  among 
the  Eskimo  tribes  of  the  Arctic  regions.  It  prevailed  among 
a  considerable  portion  of  the  North  American  Indian  tribes, 
and  was  almost  universal  with  the  native  races  of  Central  and 

1  CONANT,  op.  cit.,  p.  589. 

2  E.  B.  TYLOR,  Primitive  Culture,  New  York,  1889,  Vol.  I.,  p.  272.    In 
some  respects  a  scale  having  for  its  base  a  power  of  2  —  the  base  8  or  16, 
for  instance,  —  is  superior  to  the  duodecimal,  but  it  has  the  disadvantage 
of  not  being  divisible  by  3.     See  W.  W.  JOHNSON,  "  Octonary  Numera- 
tion," Bull.  N.  Y.  Math.  Soc.,  1891,  Vol.  I.,  pp.  1-6. 

8  TYLOR,  op.  cit.,  Vol.  I.,  p.  262. 


4  A   HISTORY    OF   MATHEMATICS 

South  America."  l  This  system  was  used  also  by  many  of  the 
North  Siberian  and  African  tribes.  Traces  of  it  are  found  in 
the  languages  of  peoples  who  now  use  the  decimal  scale ;  for 
example,  in  Homeric  Greek.  The  Roman  notation  reveals 
traces  of  it;  viz.,  I,  II, ...  V,  VI,  •••  X,  XI, ...  XV,  etc. 

It  is  curious  that  the  quinary  should  so  frequently  merge 
into  the  vigesimal  scale ;  that  savages  should  have  passed  from 
the  number  of  fingers  on  one  hand  as  an  upper  unit  or  a  stop- 
ping-place, to  the  total  number  of  fingers  and  toes  as  an  upper 
unit  or  resting-point.  The  vigesimal  system  is  less  common 
than  the  quinary,  but,  like  it,  is  never  found  entirely  pure.  In 
this  the  first  four  units  are  20,  400,  8000,  160,000,  and  special 
words  for  these  numbers  are  actually  found  among  the  Mayas 
of  Yucatan.  The  transition  from  quinary  to  vigesimal  is 
shown  in  the  Aztec  system,  which  may  be  represented  thus, 
1,2,3,  4,  5,  5  +  1,  ...  10,  10  +  1,  ...  10  +  5,10+5  +  1,  ...  20, 
20  +  1,  ...  20+10,  20  +  10  +  1,  ...  40,  etc.2  Special  words 
occur  here  for  the  numbers  1,  2,  3,  4,  5,  10,  20,  40,  etc.  The 
vigesimal  system  flourished  in  America,  but  was  rare  in  the 
Old  World.  Celtic  remnants  of  one  occur  in  the  French  words 
quatre-vingts  (4  x  20  or  80),  six-vingts  (6  x  20  or  120),  quinze- 
vingts  (15  x  20  or  300).  Note  also  the  English  word  score  in 
such  expressions  as  three-score  years  and  ten. 

Of  the  three  systems  based  on  human  anatomy,  the  decimal 
system  is  the  most  prevalent,  so  prevalent,  in  fact,  that  accord- 
ing to  ancient  tradition  it  was  used  by  all  the  races  of  the 
world.  It  is  only  within  the  last  few  centuries  that  the  other 

1  CONANT,  op.  cit.,  p.  592.      For  further  information  see  also  POTT, 
Die  quinare  und  mgesimale  Zdhlmethode  bei  Volkern  aller  Welttheile, 
Halle,  1847  ;  POTT,  Die  Sprachverschiedenheit  in  Europa  an  den  Zahl- 
wortern  nachgewiesen,  sowie  die  quinare  und  mgesimale  Zahlmethode, 
Halle,  1868. 

2  TYLOK,  op.  cit.,  Vol.  I.,  p.  262. 


NUMBER-SYSTEMS    AXD   NTJMEBALS  5 

two  systems  have  been  found  in  use  among  previously  unknown 
tribes.1  The  decimal  scale  was  used  in  North  America  by 
the  greater  number  of  Indian  tribes,  but  in  South  America  it 
was  rare. 

In  the  construction  of  the  decimal  system,  10  was  suggested 
by  the  number  of  fingers  as  the  first  stopping-place  in  count- 
ing, and  as  the  first  higher  unit.  Any  number  between  10 
and  100  was  pronounced  according  to  the  plan  6(10)  +  a(l), 
a  and  b  being  integers  less  than  10.  But  the  number  110 
might  be  expressed  in  two  ways,  (1)  as  10  x  10  +  10,  (2)  as 
11  x  10.  The  latter  method  would  not  seem  unnatural. 
Why  not  imitate  eighty,  ninety,  and  say  eleventy,  instead  of 
hundred  and  ten  ?  But  upon  this  choice  between  10  x  10  -f  10 
and  11  x  10  hinges  the  systematic  construction  of  the  number 
system.2  Good  luck  led  all  nations  which  developed  the 
decimal  system  to  the  choice  of  the  former;3  the  unit  10 
being  here  treated  in  a  manner  similar  to  the  treatment  of  the 
lower  unit  1  in  expressing  numbers  below  100.  Any  num- 
ber between  100  and  1000  was  designated  c(10)2  +  6(10)  +  a, 
a,  b,  c  representing  integers  less  than  10.  Similarly  for  num- 
bers below  10,000,  d(10)3  +  c(10)2  -f  ft(lO)1  +  a(10)°;  and  simi- 
larly for  still  higher  numbers. 

Proceeding  to  describe  the  notations  of  numbers,  we 
begin  with  the  Babylonian.  Cuneiform  writing,  as  also  the 
accompanying  notation  of  numbers,  was  probably  invented 

1  CONANT,  op.  cit.,  p.  588. 

2  HERMANN   HANKEE,  Zur  Geschichte  der  Mathematik  in  Alterthum 
und  Mittelalter,  Leipzig,  1874,  p.  11.   Hereafter  we  shall  cite  .this  brilliant 
work  as  HANKEL. 

8  In  this  connection  read  also  MORITZ  CANTOR,  Vorlesungen  uber 
Geschichte  der  Mathematik,  Vol.  I.  (Third  Edition),  Leipzig,  1907,  pp.  6 
and  7.  This  history,  by  the  prince  of  mathematical  historians  of  the 
nineteenth  century,  is  in  four  volumes,  and  will  be  cited  hereafter  as 
CANTOR. 


6  A   HISTORY   OF   MATHEMATICS 

by  the  early  Sumerians.  A  vertical  wedge  Y  stood  for 
one,  while  ^  and  y^-  signified  10  and  100,  respectively. 
In  case  of  numbers  below  100,  the  values  of  the  separate  sym- 
bols were  added.  Thus,  <?T  for  23,  <  <  <  for  30.  The 

signs  of  higher  value  are  written  on  the  left  of  those  of  lower 
value.  But  in  writing  the  hundreds  a  smaller  symbol  was 
placed  before  that  for  100  and  was  multiplied  into  100.  Thus, 

<  ]f  »-  signified  10  x  100  or  1000.  Taking  this  for  a  new 
unit,  ^  ^  If  -^-  was  interpreted,  not  as  20  x  100,  but  as 
10  x  1000.  In  this  notation  no  numbers  have  been  found  as 
large  as  a  million.1  The  principles  applied  in  this  notation 
are  the  additive  and  the  multiplicative.  Besides  this  the 
Babylonians  had  another,  the  sexagesimal  notation,  to  be 
noticed  later. 

An  insight  into  Egyptian  methods  of  notation  was  obtained 
through  the  deciphering  of  the  hieroglyphics  by  Champollion, 

Young,  and  others.  The  numerals  are  I  (1),  H  (10),  (J  (100), 
§  (1000),  |[  (10,000),  ^  (100,000),  X  (1,000,000),  O 
(10,000,000).  The  sign  for  one  represents  a  vertical  staff; 
that  for  10,000,  a  pointing  finger ;  that  for  100,000,  a  burbot ; 
that  for  1,000,000,  a  man  in  astonishment.  No  certainty  has 
been  reached  regarding  the  significance  of  the  other  symbols. 
These  numerals  like  the  other  hieroglyphic  signs  were  plainly 
pictures  of  animals  or  objects  familiar  to  the  Egyptians, 
which  in  some  way  suggested  the  idea  to  be  conveyed.  They 
are  excellent  examples  of  picture-writing.  The  principle  in- 
volved in  'the  Egyptian  notation  was  the  additive  throughout. 

Thus,   /D  A  |    would  be  111. 

1  For  fuller  treatment  see  MORITZ  CANTOR,  Mathematische  Beitrdge 
zum  Kulturleben  der  Volker,  Halle,  1863,  pp.  22-38. 


NUMBER-SYSTEMS    AND   NUMERALS  7 

Hieroglyphics  are  found  on  monuments,  obelisks,  and  walls 
of  temples.  Besides  these  the  Egyptians  had  hieratic  and 
demotic  writings,  both  supposed  to  be  degenerated  forms  of 
hieroglyphics,  such  as  would  be  likely  to  evolve  through  pro- 
longed use  and  attempts  at  rapid  writing.  The  following  are 
hieratic  signs  :  l 


10       20      30       40       50     60      70       80      90 


100      200        1000         9000 


Since  there  are  more  hieratic  symbols  than  hieroglyphic, 
numbers  could  be  written  more  concisely  in  the  former.  The 
additive  principle  rules  in  both,  and  the  symbols  for  larger 
values  always  precede  those  for  smaller  values. 

About  the  time  of  Solon,  the  Greeks  used  the  initial  letters 
of  the  numeral  adjectives  to  represent  numbers.  These  signs 
are  often  called  Herodianic  signs  (after  Herodianus,  a  Byzan- 
tine grammarian  of  about  200  A.D.,  who  describes  them). 
They  are  also  called  Attic,  because  they  occur  frequently  in 
Athenian  inscriptions.  The  Phoenicians,  Syrians,  and  Hebrews 
possessed  at  this  time  alphabets  and  the  two  latter  used  letters 
of  the  alphabet  to  designate  numbers.  The  Greeks  began  to 
adopt  the  same  course  about  500  B.C.  The  letters  of  the  Greek 
alphabet,  together  with  three  antique  letters,  r,  9,  7*),  and  the 

1  CANTOR,  Vol.  L,  pp.  84  and  85.  The  hieratic  numerals  are  taken 
from  Cantor's  table  at  the  end  of  the  volume  published  in  1894. 


8  A    HISTORY    OF    MATHEMATICS 

symbol  M,  were  used  for  numbers.  For  the  numbers  1-9  they 
wrote  a,  /?,  y,  3,  e,  5,  £,  77,  0 ;  for  the  tens  10-90,  i,  K,  X  ,  /x,  v,  £, 
o,  ir,  9 ,  for  the  hundreds  100-900,  p,  o-,  T,  v,  <£,  x>  ^  w>  7D  5  ^ or 
the  thousands  they  wrote  ,a,  t(3,  ,y,  ,8,  ,e,  etc.;  for  10,000,  M; 
for  20,000,  M;  for  30,000,  ]fr,  etc.  The  change  from  Attic  to 
alphabetic  numerals  was  decidedly  for  the  worse,  as  the 
former  were  less  burdensome  to  the  memory.  In  Greek  gram- 
mars we  often  find  it  stated  that  alphabetic  numerals  were 
marked  with  an  accent  to  distinguish  them  from  words,  but 
this  was  not  commonly  the  case ;  a  horizontal  line  drawn  over 
the  number  usually  answered  this  purpose,  while  the  accent 
generally  indicated  a  unit-fraction,  thus  8'  =  \.1  The  Greeks 
applied  to  their  numerals  the  additive  and,  in  cases  like  M 
for  50,000,  also  the  multiplicative  principle. 

In  the  Eoman  notation  we  have,  besides  the  additive,  the 
principle  of  subtraction.  If  a  letter  is  placed  before  another 
of  greater  value,  the  former  is  to  be  subtracted  from  the  latter. 
Thus,  IV  =  4,  while  VI  =  6.  Though  this  principle  has  not 
been  found  in  any  other  notation,  it  sometimes  occurs  in  numer- 
ation. Thus  in  Latin  duodeviginti  =  2  from  20,  or  18.2  The 
Eoman  numerals  are  supposed  to  be  of  Etruscan  origin. 

Thus,  in  the  Babylonian,  Egyptian,  Greek,  Eoman,  and 
other  decimal  notations  of  antiquity,  numbers  are  expressed 
by  means  of  a  few  signs,  these  symbols  being  combined  by 
addition  alone,  or  by  addition  together  with  multiplication 
or  subtraction.  But  in  none  of  these  decimal  systems  do  we 
find  the  all-important  principle  of  position  or  principle  of 

1  DR.  G.  FRIEDLEIN,  Die  Zahlzeichen  und  das  Elementare  Eechnen 
der  Griechen  und  Romer,  Erlangen,  1869,  p.  13.      The  work  will  be  cited 
after  this  as  FRIEDLEIN.     See  also  DR.  SIEGMUND  GUNTHER  in  MULLER'S 
Handbuch    der  Klassischen    Altertumswissenschaft,   Funfter    Band,    1. 
Abteilung,  1888,  p.  9. 

2  CANTOR,  Vol.  I.,  pp.  11  and  525. 


NUMBER-SYSTEMS   AND    NUMERALS  9 

local  value,  such  as  we  have  in  the  notation  now  in  use. 
Having  missed  this  principle,  the  ancients  had  no  use  for  a 
symbol  to  represent  zero,  and  were  indeed  very  far  removed 
from  an  ideal  notation.  In  this  matter  even  the  Greeks  and 
Romans  failed  to  achieve  what  a  remote  nation  in  Asia,  little 
known  to  Europeans  before  the  present  century,  accomplished 
most  admirably.  But  before  we  speak  of  the  Hindus,  we 
must  speak  of  an  ancient  Babylonian  notation,  which,  strange 
to  say,  is  not  based  on  the  scale  5,  10,  or  20,  and  which, 
moreover,  came  very  near  a  full  embodiment  of  the  ideal 
principle  found  wanting  in  other  ancient  systems.  We  refer 
to  the  sexagesimal  notation. 

The  Babylonians  used  this  chiefly  in  the  construction  of 
weights  and  measures.  The  systematic  development  of  the 
sexagesimal  scale,  both  for  integers  and  fractions,  reveals  a 
high  degree  of  mathematical  insight  on  the  part  of  the  early 
Sumerians.  The  notation  has  been  found  on  two  Babylonian 
tablets.  One  of  them,  probably  dating  from  1600  or  2300  B.C., 
contains  a  list  of  square  numbers  up  to  602.  The  first  seven 
are  1,  4,  9,  16,  25,  36,  49.  We  have  next  1.4  =  82,  1.21  =  9*, 
1.40  =  102,  2.1  =  II2,  etc.  This  remains  unintelligible,  unless 
we  assume  the  scale  of  sixty,  which  makes  1.4  =  60  -f-  4, 
1.21  =  60  +  21,  etc.  The  second  tablet  records  the  magnitude 
of  the  illuminated  portion  of  the  moon's  disc  for  every  day 
from  new  to  full  moon,  the  whole  disc  being  assumed  to  con- 
sist of  240  parts.  The  illuminated  parts  during  the  first  five 
days  are  the  series  5,  10,  20,  40,  1.20(=  80).  This  reveals 
again  the  sexagesimal  scale  and  also  some  knowledge  of 
geometrical  progressions.  From  here  on  the  series  becomes  an 
arithmetical  progression,  the  numbers  from  the  fifth  to  the 
fifteenth  day  being  respectively,  1.20,  1.36,  1.52,  2.8,  2.24,  2.40, 
2.56,  3.12,  3.28,  3.44,  4.  In  this  sexagesimal  notation  we  have, 
then,  the  principle  of  local  value.  Thus,  in  1.4  (=  64),  the  1  is 


10  A   HISTORY   OF  MATHEMATICS 

made  to  stand  for  60,  the  unit  of  the  second  order,  by  virtue  of 
its  position  with  respect  to  the  4.  In  Babylonia  some  use  was 
thus  made  of  the  principle  of  position,  perhaps  2000  years 
before  the  Hindus  developed  it.  This  was  at  a  time  when 
Eomulus  and  Remus,  yea  even  Achilles,  Menelaus,  and  Helen, 
were  still  unknown  to  history  and  song.  But  the  full  develop- 
ment of  the  principle  of  position  calls  for  a  symbol  to  represent 
the  absence  of  quantity,  or  zero.  Babylonian  records  of  about 
200  B.C.  give  a  symbol  for  zero  which  denoted  the  absence  of  a 
figure,  but  apparently  was  not  used  in  calculation.  It  con- 
sisted of  two  marks  ^ .  About  130  A.D.  Ptolemy  in  his  Al- 
magest used  sexagesimal  fractions,  and  also  the  omicron  o  to 
represent  blanks  in  the  sexagesimal  numbers.  This  b  was  not 
used  as  a  regular  zero.  Hence,  the  Babylonians  had  the  prin- 
ciple of  local  value,  and  also  a  symbol  for  zero  to  indicate  the 
absence  of  a  figure,  but  did  not  use  this  zero  in  computation. 
Their  sexagesimal  fractions  were  transmitted  to  India.  v 
'  What  was  it  that  suggested  to  the  Babylonians  the  number 
sixty  as  a  base?  Cantor1  and  others  offer  the  following 
provisional  reply  :  At  first  the  Babylonians  reckoned  the  year 
at  360  days.  This  led  to  the  division  of  the  circumference  of 
a  circle  into  360  degrees,  each  degree  representing  the  daily 
part  of  the  supposed  yearly  revolution  of  the  sun  around 
the  earth.  Probably  they  knew  that  the  radius  could  be 
applied  to  the  circumference  as  a  chord  six  times,  and  that 
each  arc  thus  cut  off  contained  60  degrees.  Thus  the 
division  into  60  parts  may  have  suggested  itself.  When 
greater  precision  was  needed,  each  degree  was  divided  into  60 
equal  parts,  or  minutes.  In  this  way  the  sexagesimal  notation 
may  have  originated.  The  division  of  the  day  into  24  hours, 
and  of  the  hour  into  minutes  and  seconds  on  the  scale  of  60, 

iVol.  I.,  pp.  91-93. 


NUMBER-SYSTEMS   AND   NUMERALS  11 

is  due  to  the  Babylonians.  There  are  also  indications  of  a 
knowledge  of  sexagesimal  fractions?  such  as  were  used  later 
by  the  Greeks,  Arabs,  by  scholars  of  the  middle  ages  and  of 
even  recent  times. 

Babylonian  science  has  made  its  impress  upon  modern  civili- 
zation. Whenever  a  surveyor  copies  the  readings  from  the 
graduated  circle  on  his  theodolite,  whenever  the  modern  man 
notes  the  time  of  day,  he  is,  unconsciously  perhaps,  but 
unmistakably,  doing  homage  to  the  ancient  astronomers  on 
the  banks  of  the  Euphrates. 

The  full  development  of  our  decimal  notation  belongs  to 
comparatively  modern  times.  Decimal  notation  had  been  in 
use  for  thousands  of  years,  before  it  was  perceived  that  its 
simplicity  and  usefulness  could  be  enormously  increased 
by  the  adoption  of  the  principle  of  position.  To  the  Hin- 
dus of  the  fifth  or  sixth  century  after  Christ  we  owe  the 
full  and  systematic  development  of  the  use  of  the  zero 
and  the  principle  of  local  value.  Of  all  mathematical  dis- 
coveries, no  one  has  contributed  more  to  the  general  progress 
of  intelligence  than  this.  While  the  older  notations  served 
merely  to  record  the  answer  of  an  arithmetical  computation, 
the  Hindu  notation  (wrongly  called  the  Arabic  notation) 
assists  with  marvellous  power  in  performing  the  computation 
itself.  To  verify  this  truth,  try  to  multiply  723  by  364,  by 
first  expressing  the  numbers  in  the  Eoman  notation;  thus, 
multiply  DCCXXIII  by  CCCLXIV.  This  notation  offers 
little  or  no  help;  the  Romans  were  compelled  to  invoke 
the  aid  of  the  abacus  in  calculations  like  this. 

Very  little  is  known  concerning  the  mode  of  evolution  of  the 
Hindu  notation.  There  is  evidence  for  the  belief  that  the 
Hindu  notation  of  the  second  century,  A.D.,  did  not  include 

i  CANTOR,  Vol.  L,  p.  31. 


12  A   HISTORY   OF   MATHEMATICS 

the  zero  nor  the  principle  of  local  value.  On  the  island  of 
Ceylon  a  notation  resembling  the  Hindu,  but  without  the  zero, 
has  been  preserved.  It  is  known  that  Buddhism  and  Indian 
culture  were  transplanted  thither  about  the  third  century  and 
there  remained  stationary.  It  is  highly  probable,  then,  that 
the  notation  of  Ceylon  is  the  old  imperfect  Hindu  system. 
Besides  signs  for  1-9,  the  Ceylon  notation  has  symbols  for 
each  of  the  tens  and  for  100  and  1000.  Thus  the  number  7685 
would  have  been  written  with  six  symbols,  designating  respec- 
tively the  numbers  7,  1000,  6,  100,  80,  5.  These  so-called 
Singhalesian  signs  are  supposed  originally  to  have  been,  like 
the  old  Hindu  numerals,  the  initial  letters  of  the  corre- 
sponding numeral  adjectives.1  Unlike  the  English,  the  first 
nine  Sanscrit  numeral  adjectives  have  each  a  different  begin- 
ning, thereby  excluding  ambiguity.  In  course  of  centuries  the 
forms  of  the  Hindu  letters  altered  materially,  but  the  letters 
that  seem  to  resemble  most  closely  the  apices  of  Boethius  and 
the  West- Arabic  numerals  (which  we  shall  encounter  later)  are 
the  letters  of  the  second  century. 

The  earliest  known  form  of  the  Indian  symbol  for  zero  was 
that  of  a  dot,  which  was  used  in  inscriptions  and  manuscripts 
in  order  to  mark  a  blank.  This  early  use  resembles  the  still 
earlier  use  made  of  symbols  for  zero  by  the  Babylonians  and 
by  Ptolemy.  Probably  Aryabhatta  in  the  fifth  century  A.D. 
knew  our  zero.  The  earliest  undoubted  occurrence  of  our 
zero  in  India  is  in  876  A.D. 

The  Hindus  possessed  several  different  modes  of  designating 
numbers.  They  sometimes  found  it  convenient  to  use  a  symbolic 
system  of  position,  in  which  1  might  be  expressed  by  "  moon  " 
or  "  earth,"  2  by  "  eye/'  etc.  In  the  Suryctrsiddhanta  2  (a  text- 


1  CANTOR,  Vol.  I.,  pp.  603-607. 

2  Translated  by   E.    BURGESS,  and  annotated  by  W.   D.  WHITNEY, 
New  Haven,  1860,  p.  3. 


NUMBER-SYSTEMS   AND   NUMERALS  13 

book  of  Hindu  astronomy)  the  number  1577917828  is  thus 
given  :  Vasu  (a  class  of  deities  8  in  number)  — two  —  eight — 
mountain  (the  7  mythical  chains  of  mountains)  —  form  — 
figure  (the  9  digits)  —  seven  —  mountain  —  lunar  days  (of 
which  there  are  15  in  the  half -month).  This  notation  is  cer- 
tainly interesting.  It  seems  to  have  been  applied  as  a  memoria 
technica  in  order  to  record  dates  and  numbers.  Such  a  selec- 
tion of  synonyms  made  it  much  easier  to  draw  up  phrases  or 
obscure  verses  for  artificial  memory.  To  a  limited  degree  this 
idea  may  perhaps  be  advantageously  applied  by  the  teacher  in 
the  schoolroom. 

The  Hindu  notation,  in  its  developed  form,  reached  Europe 
during  the  twelfth  century.  It  was  transmitted  to  the  Occi- 
dent through  the  Arabs,  hence  the  name  "  Arabic  notation." 
No  blame  attaches  to  the  Arabs  for  this  pseudo-name;  they 
always  acknowledged  the  notation  as  an  inheritance  from 
India.  During  the  1000  years  preceding  1200  A.D.,  the  Hindu 
numerals  and  notation,  while  in  the  various  stages  of  evolu- 
tion, were  carried  from  country  to  country.  Exactly  what 
these  migrations  were,  is  a  problem  of  extreme  difficulty. 
Not  even  the  authorship  of  the  letters  of  Junius  has  produced 
so  much  discussion.1  The  facts  to  be  explained  and  harmon- 
ized are  as  follows : 

1.  When,  toward  the  close  of  the  last  century,  scholars 
gradually  became  convinced  that  our  numerals  were  not  of 
Arabic,  but  of  Hindu  origin,  the  belief  was  widespread  that 
the  Arabic  and  Hindu  numerals  were  essentially  identical  in 
form.  Great  was  the  surprise  when  a  set  of  Arabic  numerals, 
the  so-called  Gubar-numeralsy  was  discovered,  some  of  which 

1  Consult  TREUTLEIN,  Geschichte  unserer  Zahlzeichen  und  Entwicke- 
lung  der  Ansichten  iiber  dieselbe,  Karlsruhe,  1875  ;  SIEGMUND  GUNTHER, 
Ziele  und  Eesultate  der  neueren  mathematisch-historischen  Forschung, 
Erlangen,  1876,  note  17. 


14  A  HISTORY  OF  MATHEMATICS 

bore  no  resemblance  whatever  to  the  modern  Hindu  characters, 
called  Devanagari-numerals. 

2.  Closer  research  showed  that  the  numerals  of  the  Arabs  of 
Bagdad  differed  from  those  of  the  Arabs  at  Cordova,  and  this 
to  such  an  extent  that  it  was  difficult  to  believe  the  westerners 
received  the  digits  directly  from  their  eastern  neighbours.    The 
West-Arabic  symbols  were  the  Gubar-numerals  mentioned  above. 
The  Arabic  digits  can  be  traced  back  to  the  tenth  century. 

3.  The  East  and  the  West  Arabs  both  assigned  to  the  num- 
erals a  Hindu  origin.    "  Gubar-numerals  "  are  "  dust-numerals/' 
in  memory  of  the  Brahmin  practice  of  reckoning  on  tablets 
strewn  with  dust  or  sand. 

4.  Not  less  startling  was  the  fact  that  both  sets  of  Arabic 
numerals  resembled  the  apices  of  Boethius  much  more  closely 
than  the  modern  Devanagari-numerals.     The  Gubar-numerals 
in  particular  bore  a  striking  resemblance  to  the  apices.     But 
what  are  the  apices  ?      Boethius,   a  Roman  writer   of  the 
sixth  century,  wrote  a  geometry,  in  which  he  speaks  of  an 
abacus,  which  he  attributes  to  the  Pythagoreans.      Instead  of 
following  the  ancient  practice  of  using  pebbles  on  the  abacus, 
he  employed  the  apices,  which  were   probably  small  cones. 
Upon  each  of  these  was  drawn  one  of  the  nine  digits,  now 
called  "  apices."     These  digits  occur  again  in  the  body  of  the 
text.1     Boethius  gives  no  symbol  for  zero. 

Need  we  marvel  that,  in  attempting  to  harmonize  these 
apparently  incongruous  facts,  scholars  for  a  long  time  failed 
to  agree  on  an  explanation  of  the  strange  metamorphoses  of 
the  numerals,  or  the  course  of  their  fleeting  footsteps,  as  they 
migrated  from  land  to  land  ? 

The  explanation  most  favourably  received  is  that  of  Woepcke.8 

1  See  FRIEDLEIN'S  edition  of  BOETHIUS,  Leipzig,  1867,  p.  397. 

2  See  Journal  Asiatique,  1st  half-year,  1863,  pp.  69-79  and  514-529. 
See  also  CANTOR,  Vol.  I.,  p.  711. 


NUMBER-SYSTEMS   AND   NUMERALS  15 

1.  The  Hindus  possessed  the  nine  numerals  without  the  zero, 
as  early  as  the  second  century  after  Christ.     It  is  known  that 
about  that  time  a  lively  commercial  intercourse  was  carried 
on  between  India  and  Rome,  by  way  of  Alexandria.     There 
arose  an  interchange  of  ideas  as  well  as  of  merchandise.     The 
Hindus  caught  glimpses  of  Greek  thought,  and  the  Alexan- 
drians received  ideas  on  philosophy  and  science  from  the  East. 

2.  The  nine  numerals,  without  the  zero,  thus  found  their 
way  to  Alexandria,  where  they  may  have  attracted  the  atten- 
tion of  the  Neo-Pythagoreans.     From  Alexandria  they  spread 
to  Rome,  thence   to  Spain  and  the  western  part  of  Africa. 
[While  the  geometry  of  Boethius  (unless  the  passage  relating 
to  the  apices  be  considered  an  interpolation  made  five  or  six 
centuries  after  Boethius)  proves  the  presence  of  the  digits  in 
Eome  in  the  fifth  century,  it  must  be  remarked  against  this 
part  of  Woepcke's  theory,  that  he  possesses  no  satisfactory 
evidence  that  they  were  known  in  Alexandria  in  the  second 
or  third  century.] 

3.  Between  the  second  and  the  eighth  centuries  the  nine 
characters  in  India  underwent  changes  in  shape.    A  prominent 
Arabic  writer,  Albirinni  (died  1038),  who  was  in  India  dur- 
ing many  years,  remarks  that  the  shape  of   Hindu  numerals 
and  letters  differed  in  different  localities  and  that  when  (in  the 
eighth  century)  the  Hindu  notation  was  transmitted  to  the 
Arabs,  the  latter  selected  from  the  various  forms  the  most 
suitable.     But  before  the  East  Arabs  thus  received  the  nota- 
tion, it  had  been  perfected  by  the  invention  of  the  zero  and 
the  application  of  the  principle  of  position. 

4.  Perceiving  the  great  utility  of  that  Columbus-egg,  the 
zero,  the  West  Arabs  borrowed  this  epoch-making  symbol  from 
those  in  the  East,  but  retained  the  old  forms  of  the  nine  num- 
erals, which  they  had  previously  received  from  Rome.     The 
reason  for  this  retention  may  have  been  a  disinclination  to 


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NUMBER-SYSTEMS   AND   NUMERALS  17 

unnecessary  change,  coupled,  perhaps,  with  a  desire  to  be  con- 
trary to  their  political  enemies  in  the  East. 

5.  The  West  Arabs  remembered  the  Hindu  origin  of  the 
old  forms,  and  called  them  "  Gubar "  or  "  dust "  numerals. 

6.  After  the  eighth  century  the  numerals  in  India  under- 
went further  changes,  and  assumed  the  greatly  modified  forms 
of  the  modern  Devanagari-numerals. 

The  controversy  regarding  the  origin  and  transmission  of 
our  numerals  has  engaged  many  minds.  Search  for  infor- 
mation has  led  to  the  close  consideration  of  intellectual, 
commercial,  and  political  conditions  among  the  Hindus,  Alex- 
andrian Greeks,  Romans,  and  particularly  among  the  East  and 
West  Arabs.  We  have  here  an  excellent  illustration  of  how 
mathematico-historical  questions  may  give  great  stimulus  to 
the  study  of  the  history  of  civilization,  and  may  throw  new 
light  upon  it.1 

In  the  history  of  the  art  of  counting,  the  teacher  finds  em- 
phasized certain  pedagogical  precepts.  We  have  seen  how 
universal  has  been  the  practice  of  counting  on  fingers.  In 
place  of  fingers,  groups  of  other  objects  were  frequently 
chosen,  "as  when  South  Sea  Islanders  count  with  cocoanut 
stalks,  putting  a  little  one  aside  every  time  they  count  to  10, 
and  a  large  one  when  they  come  to  100,  or  when  African 
negroes  reckon  with  pebbles  or  nuts,  and  every  time  they  come^ 
to  5  put  them  aside  in  a  little  heap." 2  The  abstract  notion  of 
number  is  here  attained  through  the  agency  of  concrete  objects. 
The  arithmetical  truths  that  2  +  1  =  3,  etc.,  are  here  called 
out  by  experience  in  the  manipulation  of  things.  As  we  shall 
see  later,  the  counting  by  groups  of  objects  in  early  times  led 
to  the  invention  of  the  abacus,  which  is  still  a  valuable  school 
instrument.  The  earliest  arithmetical  knowledge  of  a  child 

1  GUNTHER,  Op.   Cft.,  p.  13.  2  TYLOR,  Op.   Clt.,  Vol.  I.,  p.  270. 


18  A    HISTORY    OF   MATHEMATICS 

should,  therefore,  be  made  to  grow  out  of  his  experience  with 
different  groups  of.  objects  ;  never  should  he  be  taught  count- 
ing by  being  removed  from  his  toys,  and  (practically  with 
his  eyes  closed)  made  to  memorize  the  abstract  statements 
l-fl  =  2,  2+1  =  3,  etc.  Primitive  counting  in  its  mode  of 
evolution  emphasizes  the  value  of  object-teaching. 

Recent  research1  seems  to  indicate  that  the  principle  of 
local  value  and  the  use  of  the  zero  received  systematic  and  full 
development  in  America  several  centuries  earlier  than  in  Asia. 
The  credit  of  this  achievement  belongs  to  the  Maya  of  Central 
America.  About  the  beginning  of  the  Christian  era  the  Maya 
possessed  a  fully  developed  number-system  and  chronology. 
Theirs  was  not  the  decimal,  but  the  vigesimal  notation.  Their 
system,  as  exhibited  in  their  codices,  was  based  on  the  scale 
of  20,  except  in  the  third  position.  That  is,  20  units  of  the 
lowest  order  (kins,  or  days)  make  1  unit  of  next  higher  order 
(uinals,  or  20  days),  18  uinals  make  1  tun  (360  days,  the  Maya 
official  year),  20  tuns  make  1  katun  (7200  days),  20  katuns 
make  1  cycle,  and  20  cycles  make  1  great  cycle.  The  numbers 
1-19  are  expressed  by  bars  and  dots.  Each  bar  stands  for  5, 
each  dot  for  1.  Thus,  •  •  is  2,  _^  is  6.  The  zero  is  repre- 
sented by  a  symbol  that  looks  roughly  like  a  half-closed  eye. 
Accordingly,  37  was  expressed  by  the  symbols  for  17  (three 
bars  and  two  dots)  in  the  kin  place,  and  one  dot  representing 
20  in  the  uinal  place,  higher  up.  The  numbers  were  written 
vertically.  To  write  360,  the  Maya  drew  two  zeros,  one  above 
the  other,  with  a  dot  in  third  place,  above  the  zeros.  The 
largest  number  found  in  the  codices  is  12,489,781. 

1  S.  G.  MORLBY,  Introduction  to  the  Study  of  the  Maya  Hieroglyphs, 
Washington,  1915. 


ARITHMETIC  AND  ALGEBRA 


EGYPT 

THE  most  ancient  mathematical  handbook  known  to  our 
time  is  a  papyrus  included  in  the  Rhind  collection  of  the 
British  Museum.  This  interesting  hieratic  document,  de- 
scribed by  Birch  in  1868,  and  translated  by  Eisenlohr1  in 
1877,  was  written  by  an  Egyptian,  Ahmes  by  name,  in  the 
reign  of  Ra-a-us,  some  time  between  1700  and  2000  B.C.  It  is 
entitled :  "  Directions  for  obtaining  the  knowledge  of  all  dark 
things."  It  claims  to  be  founded  on  older  documents  pre- 
pared in  the  time  of  King  [Ra-en-m]at.  Unless  specialists 
are  in  error  regarding  the  name  of  this  last  king,  Ra-en-mat, 
[i.e.  Amenemhat  III.],  whose  name  is  not  legible  in  the 
papyrus,  it  follows  that  the  original  is  many  centuries  older 
than  the  copy  made  by  Ahmes.  The  Ahmes  papyrus,  there- 
fore, gives  us  an  idea  of  Egyptian  geometry,  arithmetic,  and 
algebra  as  it  existed  certainly  as  early  as  1700  B.C.,  and 
possibly  as  early  as  3000  B.C.  While  it  does  not  disclose  as 
extensive  mathematical  knowledge  as  one  might  expect  to  find 
among  the  builders  of  the  pyramids,  it  nevertheless  shows  in 

1  A.  EISENLOHR,  Ein  mathematisches  Handbuch  der  alien  Aegypter 
(Papyrus  Rhind  des  British  Museum'),  Leipzig,  1877;  2d  ed.  1891. 
See  also  CANTOR,  I.,  pp.  59-84,  and  JAMES  Gow,  A  Short  History  of  Greek 
Mathematics,  Cambridge,  1884,  pp.  16-19.  The  last  work  will  be  cited 
hereafter  as  Gow. 

19 


20  A   HISTORY   OF   MATHEMATICS 

several  particulars  a  remarkably  advanced  state  of  mathe- 
matics at  the  time  when  Abraham  visited  Egypt. 

From  the  Ahmes  papyrus  we  infer  that  the  Egyptians  knew 
nothing  of  theoretical  results.  It  contains  no  theorems,  and 
hardly  any  general  rules  of  procedure.  In  most  cases  the 
writer  treats  in  succession  several  problems  of  the  same  kind. 
From  them  it  would  be  easy,  by  induction,  to  obtain  general 
rules,  but  this  is  not  done.  When  we  remember  that  only  one 
hundred  years  ago  it  was  the  practice  of  many  English  arith- 
metical writers  to  postpone  the  discussion  of  fractions  to  the 
end  of  their  books,  it  is  surprising  to  find  that  this  hand-book 
of  4000  years  ago  begins  with  exercises  in  fractions,  and  pays 
but  little  attention  to  whole  numbers.  Gow  probably  is  right 
in  his  conjecture  that  Ahmes  wrote  for  the  elite  mathematicians 
of  his  day. 

While  fractions  occur  in  the  oldest  mathematical  records 
which  have  been  found,  the  ancients  nevertheless  attained 
little  proficiency  in  them.  Evidently  this  subject  was  one  of 
great  difficulty.  Simultaneous  changes  in  both  numerator  and 
denominator  were  usually  avoided.  Fractions  are  found  among 
the  Babylonians.  Not  only  had  they  sexagesimal  divisions  of 
weights  and  measures,  but  also  sexagesimal  fractions.1  These 
fractions  had  a  constant  denominator  (60),  and  were  indicated 
by  writing  the  numerator  a  little  to  the  right  of  the  ordinary 
position  for  a  word  or  number,  the  denominator  being  under- 
stood. We  shall  see  that  the  Eomans  likewise  usually  kept 
the  denominators  constant,  but  equal  to  12.  The  Egyptians 
and  Greeks,  on  the  other  hand,  kept  the  numerators  constant 
and  dealt  with  variable  denominators.  Ahmes  confines  him- 
self to  fractions  of  a  special  class,  namely  unit-fractions,  having 
unity  for  their  numerators.  A  fraction  was  designated  by 

1  CANTOR,  Vol.  I.,  pp.  23,  31. 


EGYPT  21 

writing  the  denominator  and  then  placing  over  it  either  a  dot 
or  a  symbol,  called  ro.  Fractional  values  which  could  not  be 
expressed  by  any  one  unit-fraction  were  represented  by  the  sum 
of  two  or  more  of  them.  Thus  he  wrote  ^  -Jg-  in  place  of  -J. 
It  is  curious  to  observe  that  while  Ahmes  knows  f  to  equal  1  J-, 
he  makes  an  exception  in  this  case,  adopts  a  special  symbol 
for  J,  and  allows  it  to  appear  often  among  the  unit-fractions.1 
A  fundamental  problem  in  Ahmes's  treatment  of  fractions  was, 
how  to  find  the  unit-fractions,  the  sum  of  which  represents  a 
given  fractional  value.  This  was  done  by  aid  of  a  table,  given 

o 

in  the  papyrus,  in  which  all   fractions  of  the  form  — 

2  n  -f- 1 

(n  designating  successively  all  integers  up  to  49)  are  reduced  to 
the  sum  of  unit-fractions.  Thus,  ^  =  ^  A>  A  =  T2  A  rk 
•g-J-y.  With  aid  of  this  table  Ahmes  could  work  out  problems 
like  these,  "  Divide  2  by  3,"  "  Divide  2  by  17,"  etc.  Ahmes 
nowhere  states  why  he  confines  himself  to  2  as  the  numer- 
ator, nor  does  he  inform  us  how,  when,  and  by  whom  the  table 
was  constructed.  It  is  plain,  however,  that  by  the  use  of 
this  table  any  fraction  whose  denominator  is  odd  and  less  than 
100,  can  be  represented  as  the  sum  of  unit-fractions.  The 
division  of  5  by  21  may  have  been  accomplished  as  follows : 
5  =  1+2  +  2.  From  the  table  we  get  T2T  =  ^  ^.  Then 

*  =  *  +  «*  A)  +  (A  A)=A  +  (T24  422)  =  A  I  *  =  t  A 
—  T  Tt  A-  -ft  mav  be  remarked  that  there  are  many  ways  of 
breaking  up  a  fraction  into  unit-fractions,  but  Ahmes  invariably 
gives  only  one.  Contrary  to  his  usual  practice,  he  gives  a 
general  rule  for  multiplying  a  fraction  by  f .  He  says :  "  If 
you  are  asked,  what  is  f  of  |-,  take  the  double  and  the  sixfold ; 
that  is  f  of  it.  One  must  proceed  likewise  for  any  other  frac- 
tion." As  only  the  denominator  was  written  down,  he  means 

i  CANTOR,  Vol.  I.,  p.  61. 


22  A  HISTORY  OF  MATHEMATICS 

by  the  "  double  "  and  "  sixfold,"  double  and  sixfold  the  denom- 
inator.    Since  }  =  •£••£•,  the  rule  simply  means 

211  1 

3X5  =  2  x5  +  6x5 

His  statement  "likewise  for  any  other  fraction"  appears  to 
mean l 

Sx-i-l+i 

3     a     2a     6a 

The  papyrus  contains  17  examples  which  show  by  what  a 
fraction  or  a  mixed  number  must  be  multiplied,  or  what  must 
be  added  to  it,  to  obtain  a  given  value.  The  method  consists 
in  the  reduction  of  the  given  fractions  to  a  common  denomina- 
tor. Strange  to  say,  the  latter  is  not  always  chosen  so  as  to 
be  exactly  divisible  by  all  the  given  denominators.  Ahmes 
gives  the  example,  increase  \  i  y1^  -fa  ^  to  1.  The  common 
denominator  taken  appears  to  be  45,  for  the  numbers  are 
stated  as  11  J,  5^  |,  4£,  1£,  1.  The  sum  of  these  is  23  |J| 
forty-fifths.  Add  to  this  %  J^,  and  the  sum  is  f.  Add  £, 
and  we  have  1.  Hence  the  quantity  to  be  added  to  the  given 
fraction  is  ^  -J-  ^. 

By  what  must  y^-  yy^-  be  multiplied  to  give  i  ?  The  com- 
mon denominator  taken  is  28,  then  q-^  =  -^^>  7Ta  =  ™>  their 

r>  -•          qi  1"  *O        11^         <^0 

sum  =  ^.    Again  1  =  £|.     Since  2  +  1  +  i  =  3J,  take  first  ^, 

^O  O  ^O  j  qi 

then  its  half  ^,  then  half  of  that  ~^  and  we  have  ~     Hence 

A  TTT  becomes  £  on  multiplication  by  1^-  \. 

These  examples  disclose  methods  quite  foreign  to  modern 
mathematics.2  One  process,  however,  found  extensive  appli- 

1  CANTOR,  Vol.  I.,  p.  66. 

2  Cantor's  account  of  fractions  in  the  Ahmes  papyrus  suggested  to 
J.  J.  Sylvester  (then  in  Baltimore)  material  for  a  paper  "  On  a  Point  in 
the  Theory  of  Vulgar  Fractions,"  Am.  Journal  of  Mathematics,  III.,  1880, 
pp.  32  and  388. 


EGYPT  23 

cation  in  arithmetics  of  the  fifteenth  century  and  later, 
namely,  that  of  aliquot  parts,  used  largely  in  Practice.  In  the 
second  of  the  above  examples  aliquot  parts  are  taken  of  ^V 
This  process  is  seen  again  in  Ahmes's  calculations  to  verify 
the  identities  in  the  table  of  unit-fractions. 

Ahmes  then  proceeds  to  eleven  problems  leading  to  simple 
equations  with  one  unknown  quantity.  The  unknown  is  called 
hau  or  'heap.'  Symbols  are  used  to  designate  addition,  sub- 
traction, and  equality.  We  give  the  following  specimen  :  * 


Ha'  neb-f      ma-f    ro  sefex-f    hi-f     xePer-f    em  sa  sefe*  ; 

Heap  its  f  ,       its  |,        its  \,  its  whole,  it  gives          37 

i.e.  x(|         +i         +\         +  1)         =  37 

Here  A  stands  for  |,  /  for  1.  Other  unit-fractions  are 
indicated  by  writing  the  number  and  placing  dH>,  ro,  above  it. 
A  problem  resembling  the  one  just  given  reads:  "Heap,  its  •§, 

its  |,  its  T,  its  whole,  it  makes  33;"  i.e.  -x  +  -  +  --foj  =  33. 

o         2i      t 


The  solution  proceeds  as  follows  :  1  f  1  1  x  =  33.  Then  1 
is  multiplied  in  the  manner  sketched  above,  so  as  to  get  33, 
thereby  obtaining  heap  equal  to  U  ±  ^  £,  ^  rrit  tt  ^. 
Here,  then,  we  have  the  solution  of  an  algebraic  equation  ! 

In  this  Egyptian  document,  as  also  among  early  Babylonian 
records,  are  found  examples  of  arithmetical  and  geometrical 
progressions.  Ahmes  gives  an  example  :  "  Divide  100  loaves 
among  5  persons  ;  ^  of  what  the  first  three  get  is  what  the  last 
two  get.  What  is  the  difference  ?  "  2  Ahmes  gives  the  solu- 
tion: "Make  the  difference  51;  23,  171,  12,  6|,  1.  Multiply 

1  LUDWIG  MATTHIESSEN,  Grundzuge  der  Antiken  und  Modernen  Alge- 
bra der  litteralen  Gleichungen,  Leipzig,  1878,  p.  269.     Hereafter  we  cite 
this  work  as  MATTHIESSEN. 

2  CANTOR,  Vol.  I.,  p.  78. 


24  A    HISTORY   OF   MATHEMATICS 

by  If;  38£,  29  J,  20,  10  f  1,  If."  How  did  Ahmes  come  upon 
5  J  ?  Perhaps  thus : *  Let  a  and  —  d  be  the  first  term  and 
the  difference  in  the  required  arithmetical  progression,  then 
i  [a  +  (a  —  d)  +  (a  —  2d)]  =  (a  —  3d)  +  (a  —  4cT),  whence  d  = 
5^(a— 4d),  i.e.  the  difference  d  is  5^-  times  the  last  term. 
Assuming  the  last  term  =  1,  he  gets  his  first  progression.  The 
sum  is  60,  but  should  be  100 ;  hence  multiply  by  If,  for 
60  x  If  =  100.  We  have  here  a  method  of  solution  which 
appears  again  later  among  the  Hindus,  Arabs,  and  modern 
Europeans  —  the  method  of  false  position.  It  will  be  ex- 
plained more  fully  elsewhere. 

Still  more  curious  is  the  following  in  Ahmes.  He  speaks 
of  a  ladder  consisting  of  the  numbers  7,  49,  343,  2401,  16807. 
Adjacent  to  these  powers  of  7  are  the  words  picture,  cat,  mouse, 
barley,  measure.  Nothing  in  the  papyrus  gives  a  clue  to  the 
meaning  of  this,  but  Cantor  thinks  the  key  to  be  found  in  the 
following  problem  occurring  3000  years  later  in  the  liber  abaci 
(1202  A.D.)  of  Leonardo  of  Pisa :  7  old  women  go  to  Rome ; 
each  woman  has  7  mules,  each  mule  carries  7  sacks,  each  sack 
contains  7  loaves,  with  each  loaf  are  7  knives,  each  knife  is 
put  in  7  sheaths.  What  is  the  sum  total  of  all  named  ?  This 
has  suggested  the  following  wording  in  Ahmes :  7  persons 
have  each  7  cats ;  each  cat  eats  7  mice,  each  mouse  eats  7  ears 
of  barley,  from  each  ear  7  measures  of  corn  may  grow.  What 
is  the  series  arising  from  these  data,  what  the  sum  of  its 
terms  ?  Ahmes  gives  the  numbers,  also  their  sum,  19607. 
Problems  of  this  sort  may  have  been  proposed  for  amusement. 
If  the  above  interpretations  are  correct,  it  looks  as  if  "  mathe- 
matical recreations "  were  indulged  in  by  scholars  forty 
centuries  ago. 

In  the  hau-pioblems,  of  which  we  gave  one  example,  we  see 
the  beginnings  of  algebra.  So  far  as  documentary  evidence 

Vol.  I.,  p.  78. 


EGYPT  25 

goes,  arithmetic  and  algebra  are  coeval.  That  arithmetic  is 
actually  the  older  there  can  be  no  doubt.  But  mark  the 
close  relation  between  them  at  the  very  beginning  of  authentic 
history.  So  in  mathematical  teaching  there  ought  to  be  an 
intimate  union  between  the  two.  In  the  United  States  algebra 
was  for  a  long  time  set  aside,  while  extraordinary  emphasis 
was  laid  on  arithmetic.  The  readjustment  has  come.  The 
"Mathematical  Conference  of  Ten"  of  1892  voiced  the  senti- 
ment of  the  best  educators  when  it  recommended  the  earlier 
introduction  of  certain  parts  of  elementary  algebra. 

The  part  of  Ahmes's  papyrus  which  has  to  the  greatest 
degree  taxed  the  ingenuity  of  specialists  is  the  table  of  unit- 
fractions.  How  was  it  constructed  ?  Some  hold  that  it  was 
not  computed  by  any  one  person,  nor  even  in  one  single  epoch, 
and  that  the  method  of  construction  was  not  the  same  for  all 
fractions.  On  the  other  hand  Loria  thinks  he  has  discovered 
a  general  mode  by  which  this  and  similar  existing  tables  may 
have  been  calculated.1 

That  the  period  of  Ahmes  was  a  flowering  time  for  Egyptian 
mathematics  appears  from  the  fact  that  there  exist  two  other 
papyri  of  this  period,  containing  quadratic  equations  !  They 
were  found  at  Kahun,  south  of  the  pyramid  of  Illahun.  These 
documents  bear  close  resemblance  to  Ahmes's,  as  does  also  the 
Akhmim  papyrus,2  recently  discovered  at  Akhmim,  a  city  on 

1  See  the  following  articles  by  GINO  LORIA:  "Congetture  e  ricerche 
sulP  aritmetica  degli  antichi  Egiziani  "  in  Bibliotheca  Mathematica,  1892, 
pp.  97-109  ;  "Un  miovo  documento  relative  alia  logistico  greco-egiziana," 
Ibid.,  1893,  pp.  79-89;    "Studi  intorno  alia  logistica  greco-egiziana," 
Estratto  dal  Volume  XXXII  (1°  della  2°  serie)  del  Giornale  di  Mathema- 
tiche  di  Battaglini,  pp.  7-35. 

2  J.  BAILLET,  "  Le  papyrus  mathe'matique  <T  Akhmim,"  Memoires  pub- 
lies  par  les  membres  de  la  mission  archeologique  franqaise  au   Caire, 
T.  IX.,  lr  fascicule,  Paris,  1892,  pp.  1-88.     See  also  LORIA'S  papers  in 
the  preceding  note,  and  CANTOR,  Vol.  I.,  1907,  pp.  95,  96. 


26  A   HISTORY    OF    MATHEMATICS 

the  Nile  in  Upper  Egypt.  It  is  in  Greek  and  is  supposed  to 
have  been  written  at  some  time  between  500  and  800  A.D.  Like 
his  ancient  predecessor  Ahmes,  its  author  gives  tables  of  unit- 
fractions.  It  marks  no  progress  over  the  arithmetic  of  Ahmes. 
For  more  than  two  thousand  years  Egyptian  mathematics  was 
stationary  ! 

GREECE 

In  passing  to  Greek  arithmetic  and  algebra,  we  first  observe 
that  the  early  Greeks  were  not  automaths ;  they  acknowledged 
the  Egyptian  priests  to  have  been  their  teachers.  While  in 
geometry  the  Greeks  soon  reached  a  height  undreamed  of  by 
the  Egyptian  mind,  they  contributed  hardly  anything  to  the 
art  of  calculation.  Not  until  the  golden  period  of  geometric 
discovery  had  passed  away,  do  we  find  in  Nicomachus  and 
Diophantus  substantial  contributors  to  algebra. 

Greek  mathematicians  were  in  the  habit  of  discriminating 
between  the  science  of  numbers  and  the  art  of  computation. 
The  former  they  called  arithmetica,  the  latter  logistica. 

Greek  writers  seldom  refer  to  calculation  with  alphabetic 
numerals.  Addition,  subtraction,  and  even  multiplication  were 
probably  performed  on  the  abacus.  Eutocius,  a  commentator1 
of  the  sixth  century  A.D.,  exhibits  a  great  many  multiplications, 
such  as  expert  Greek  mathematicians  of  classical  time  may 
have  used.1  While  among  the  Sophists  computation  received 
some  attention,  it  was  pronounced  a  vulgar  and  childish 
art  by  Plato,  who  cared  only  for  the  philosophy  of  arithmetic. 

Greek  writers  did  not  confine  themselves  to  unit-fractions  as 
closely  as  did  the  Egyptians.  Unit-fractions  were  designated 
by  simply  writing  the  denominator  with  a  double  accent. 

1  For  specimens  of  such  multiplications  see  CANTOR,  Beitrdge  z.  Kulturl. 
d.  Volker,  p.  393  ;  HANKEL,  p.  56  ;  Gow,  p.  50  ;  FRIEDLEIN,  p.  76  ;  my 
History  of  Mathematics,  1895,  p.  66. 


GREECE  27 

Thus,  pifi"  =  yf 3--  Other  fractions  were  usually  indicated  by 
writing  the  numerator  once  with  an  accent  and  the  denomina- 
tor twice  with  a  double  accent.  Thus,  i£W'Ka"  =  ^-f.  As 
with  the  Egyptians,  unit-fractions  in  juxtaposition  are  to  be 
added. 

Like  the  Eastern  nations,  the  Egyptians  and  Greeks  employ 
two  aids  to  computation,  the  abacus  and  finger  symbolism.  It 
is  not  known  what  the  signs  used  in  the  latter  were,  but  by  the 
study  of  ancient  statuary,  bas-reliefs,  and  paintings  this  secret 
may  yet  be  unravelled.  Of  the  abacus  there  existed  many 
forms  at  different  times  and  among  the  various  nations.  In 
all  cases  a  plane  was  divided  into  regions  and  a  pebble  or  other 
object  represented  a  different  value  in  different  regions.  We 
possess  no  detailed  information  regarding  the  Egyptian  or 
the  Greek  abacus.  Herodotus  (II.,  36)  says  that  the  Egyp- 
tians "  calculate  with  pebbles  by  moving  the  hand  from  right 
to  left,  while  the  Hellenes  move  it  from  left  to  right."  This 
indicates  a  primitive  and  instrumental  mode  of  counting  with 
aid  of  pebbles.  The  fact  that  the  hand  was  moved  towards  the 
right,  or  towards  the  left,  indicates  that  the  plane  or  board  was 
divided  by  lines  which  were  vertical,  i.e.  up  and  down,  with 
respect  to  the  computer.  lamblichus  informs  us  that  the 
abacus  of  the  Pythagoreans  was  a  board  strewn  with  dust  or 
sand.  In  that  case,  any  writing  could  be  easily  erased  by 
sprinkling  the  board  anew.  A  pebble  placed  in  the  right  hand 
space  or  column  designated  1,  if  placed  in  the  second  column 
from  the  right  10,  if  in  the  third  column  100,  etc.  Probably, 
never  more  than  nine  pebbles  were  placed  in  one  column,  for 
ten  of  them  would  equal  one  unit  of  the  next  higher  order. 
The  Egyptians  on  the  other  hand  chose  the  column  on  the 
extreme  left  as  the  place  for  units,  the  second  column  from  the 
left  designating  tens,  the  third  hundreds,  etc.  In  further 
support  of  this  description  of  the  abacus,  a  comparison  attrib- 


28  A   HISTORY   OF   MATHEMATICS 

uted  by  Diogenes  Laertius  (I.,  59)  to  Solon  is  interesting : l  "A 
person  friendly  with  tyrants  is  like  the  stone  in  computation, 
which  signifies  now  much,  now  little." 

The  abacus  appears  to  have  been  used  in  Egypt  and  Greece 
to  carry  out  the  simpler  calculation  with  integers.  The  hand- 
book of  Ahmes  with  its  treatment  of  fractions  was  presumably 
written  for  those  already  familiar  with  abacal  or  digital  reck- 
oning. Greek  mathematical  works  usually  give  the  numerical 
results  without  exhibiting  the  computation  itself.  Thus 
advanced  mathematicians  frequently  had  occasion  to  extract 
the  square  root.  In  his  Mensuration  of  the  Circle,  Archimedes 
states,  for  instance,  that  V3  <  -VW-  and  V3  >  f  ff ,  but  he 
gives  no  clue  to  his  method  of  approximation.2 

When  sexagesimal  numbers  (introduced  from  Babylonia 
into  Greece  about  the  time  of  the  Greek  geometer  Hypsicles 
and  the  Alexandrian  astronomer  Ptolemaeus)  were  used,  then 
the  mode  of  root-extraction  resembled  that  of  the  present  time. 
A  specimen  of  the  process,  as  given  by  Theon,  the  father  of 
Hypatia,  has  been  preserved.  He  finds  V45000  =  67°  4'  55". 

Archimedes  showed  how  the  Greek  system  of  numeration 
might  be  extended  so  as  to  embrace  numbers  as  large  as  you 
please.  By  the  ordinary  nomenclature  of  his  day,  numbers 
could  be  expressed  up  to  108.  Taking  this  108  as  a  unit  of 
second  order,  1016  as  one  of  the  third  order,  etc.,  the  system 
may  be  sufficiently  extended  to  enable  one  to  count  the  very 
sands.  Assuming  10,000  grains  of  sand  to  fill  the  space  of  a 
poppy-seed,  he  finds  a  number  which  would  exceed  the  number 

1  CANTOR,  Vol.  I.,  p.  132. 

2  What  the  Archimedean  and,  in  general,  the  Greek  method  of  root- 
extraction  really  was,  has  been  a  favourite  subject  of  conjecture.    See  for 
example,  H.  WEISSENBORN'S  Berechnung  des  Kreisumfanges  bei  Archi- 
medes und  Leonardo  Pisano,  Berlin,  1894.    For  bibliography  of  this  sub- 
ject, see  S.  GUNTHER,  Gesch.  d.  antiken  Naturwissenschaft  u.  Philosophic, 
p.  16. 


GREECE  29 

of  grains  in  a  sphere  whose  radius  extends  from  the  earth  to 
the  fixed  stars.  A  counterpart  of  this  interesting  speculation, 
called  the  "  sand-counter "  (arenarius),  is  found  in  a  calcula- 
tion attributed  to  Buddha,  the  Hindu  reformer,  of  the  number 
of  primary  atoms  in  a  line  one  mile  in  length,  when  the  atoms 
are  placed  one  against  another. 

The  science  of  numbers,  as  distinguished  from  the  art  of 
calculation,  commanded  the  lively  attention  of  the  Pythago- 
reans. Pythagoras  himself  had  imbibed  Egyptian  mathemat- 
ics and  mysticism.  Aside  from  the  capital  discovery  of 
irrational  quantities  (spoken  of  elsewhere)  no  very  substantial 
contribution  was  made  by  the  Pythagoreans  to  the  science  of 
numbers.  We  may  add  that  by  the  Greeks  irrationals  were 
not  classified  as  numbers.  The  Pythagoreans  sought  the 
origin  of  all  things  in  numbers ;  harmony  depended  on  musi- 
cal proportion;  the  order  and  beauty  of  the  universe  have 
their  origin  in  numbers ;  in  the  planetary  motions  they  dis- 
cerned a  wonderful  "harmony  of  the  spheres.'7  Moreover, 
some  numbers  had  extraordinary  attributes.  Thus,  one  is  the 
essence  of  things ;  four  is  the  most  perfect  number,  correspond- 
ing to  the  human  soul.  According  to  Philolaus,  5  is  the  cause 
of  colour,  6  of  cold,  7  of  mind,  health,  and  light,  8  of  love  and 
friendship.1  Even  Plato  and  Aristotle  refer  the  virtues  to 
numbers.  While  these  speculations  in  themselves  were  fan- 
tastic and  barren,  lines  of  fruitful  mathematical  inquiry  were 
suggested  by  them. 

The  Pythagoreans  classified  numbers  into  odd  and  even, 
and  observed  that  the  sum  of  the  odd  numbers  from  1  to 
2n  +  l  was  always  a  perfect  square.  Of  no  particular  value 
were  their  classifications  of  numbers  into  heteromecic,  trian- 
gular, perfect,  excessive,  defective,  amicable.2  The  Pythag- 

1  Gow,  p.  69. 

2  For  their  definitions  see  Gow,  p.  70,  or  CAJORI,  op.  cit.,  p.  68. 


30  A   HISTORY   OF   MATHEMATICS 

oreans  paid  much  attention  to  the  subject  of  proportion.  The 
quantities  a,  b,  c,  d,  were  said  to  be  in  arithmetical  proportion, 
when  a  —  b=c— d ;  in  geometrical  proportion,  when  a  :  b  =  c  :  d ; 
in  harmonic  proportion,  when  a  —  b:b  —  c=a:c;  in  musical 
proportion,  when  a  :  |(a  +  b)=2ab/(a  +  ft)  :  5.  lamblichus 
says  that  the  last  was  introduced  from  Babylon. 

The  7th,  8th,  and  9th  books  of  Euclid's  Elements  are  on  the 
science  of  number,  but  the  2d  and  10th,  though  professedly 
geometrical  and  treating  of  magnitudes,  are  applicable  to 
numbers.  Euclid  was  a  geometer  through  and  through,  and 
even  his  arithmetical  books  smack  of  geometry.  Consider,  for 
instance,  definition  21,  book  VII.1  "  Plane  and  solid  numbers 
are  similar  when  their  sides  are  proportional."  Again,  num- 
bers are  not  written  in  numerals,  nor  are  they  designated  by 
anything  like  our  modern  algebraic  notation ;  they  are  repre- 
sented by  lines.  This  symbolism  is  very  unsuggestive.  Fre- 
quently properties  which  our  notation  unmasks  at  once  could 
be  extracted  from  these  lines  only  through  a  severe  process  of 
reasoning.2 

In  the  7th  book  we  encounter  for  the  first  time  a  definition 
of  prime  numbers.  Euclid  finds  the  G.  C.  D.  of  two  numbers 
by  a  procedure  identical  with  our  method  by  division.  He 
applies  to  numbers  the  theory  of  proportion  which  in  the  5th 
book  is  developed  for  magnitudes  in  general.  The  8th  book 
deals  with  numbers  in  continued  proportion.  The  9th  book 
finishes  that  subject,  deals  with  primes,  and  contains  the  proof 
for  the  remarkable  theorem  20,  that  the  number  of  primes  is 
infinite. 

During  the  four  centuries  after  Euclid,  geometry  monopo- 
lized the  attention  of  the  Greeks  and  the  theory  of  numbers 

1  HEIBERG'S  edition,  Vol.  II.,  p.  189. 

2  G.  H.  F.  NESSELMANN,  Die  Algebra  der  Griechen,  Berlin,  1842,  p.  184. 
To  be  cited  hereafter  as  NESSELMANN. 


GREECE  31 

was  neglected.  Of  this  period  only  two  names  deserve  men- 
tion, Eratosthenes  (about  275-194  B.C.)  and  Hypsicles  (between 
200  and  100  B.C.).  To  the  latter  we  owe  researches  on  polyg- 
onal numbers  and  arithmetical  progressions.  Eratosthenes 
invented  the  celebrated  "  sieve "  for  finding  prime  numbers. 
Write  down  in  succession  all  odd  numbers  from  3  up.  By 
erasing  every  third  number  after  3,  sift  out  all  multiples  of  3  ; 
by  erasing  every  fifth  number  after  5,  sift  out  all  multiples 
of  5,  and  so  on.  The  numbers  left  after  this  sifting  are  all 
prime.  While  the  invention  of  the  "sieve"  called  for  no 
great  mental  powers,  it  is  remarkable  that  after  Eratosthenes 
no  advance  was  made  in  the  mode  of  finding  the  primes,  nor  in 
the  determination  of  the  number  of  primes  which  exist  in  the 
numerical  series  1,  2,  3,  ...w,  until  the  nineteenth  century, 
when  Gauss,  Legendre,  Dirichlet,  Eiemann,  and  Chebichev 
enriched  the  subject  with  investigations  mostly  of  great  diffi- 
culty and  complexity. 

The  study  of  arithmetic  was  revived  about  100  A.D.  by 
Nicomachus,1  a  native  of  Gerasa  (perhaps  a  town  in  Arabia) 
and  known  as  a  Pythagorean.  He  wrote  in  Greek  a  work 
entitled  Introductio  Arithmetica.  The  historical  importance  of 
this  work  is  great,  not  so  much  on  account  of  original  matter 
therein  contained,  but  because  it  is  (so  far  as  we  know)  the 
earliest  systematic  text-book  on  arithmetic,  and  because  for 
over  1000  years  it  set  the  fashion  for  the  treatment  of  this 
subject  in  Europe.  In  a  small  measure,  Nicomachus  did  for 
arithmetic  what  Euclid  did  for  geometry.  His  arithmetic  was 
as  famous  in  his  day  as  was,  later,  Adam  Kiese's  in  Germany, 
and  Cocker's  in  England.  Wishing  to  compliment  a  computer, 
Lucian  says,  "  You  reckon  like  Nicomachus  of  Gerasa." 2  The 

1  See  NESSELMANN,  pp.  191-216  ;  Gow,  pp.  88-95 ;  CANTOR,  Vol.   I., 
pp.  400-404. 

2  Quoted  by  Gow  (p.  89)  from  Philopatris,  12. 


82  A   HISTORY   OF   MATHEMATICS 

work  was  brought  out  in  a  Latin  translation  by  Appuleius 
(now  lost)  and  then  by  Boethius.  In  Boethius's  translation 
the  elementary  parts  of  the  work  were  in  high  authority  in 
Western  Europe  until  the  country  was  inv.aded  by  Hindu 
arithmetic.  Thereupon  for  several  centuries  Greek  arithmetic 
bravely  but  vainly  struggled  for  existence  against  its  immeas- 
urably superior  Indian  rival. 

The  style  of  Nicomachus  differs  essentially  from  that  of  his 
predecessors.  It  is  not  deductive,  but  inductive.  The  geo- 
metrical style  is  abandoned;  the  different  classes  of  numbers 
are  exhibited  in  actual  numerals.  The  author's  main  object  is 
classification.  Being  under  the  influence  of  philosophy  an'd 
theology,  he  sometimes  strains  a  point  to  secure  a  division  into 
groups  of  three.  Thus,  odd  numbers  are  either  "prime  and 
uncompounded,"  "compounded,"  or  "compounded  but  prime 
to  one  another."  His  nomenclature  resulting  from  this  classi- 
fication is  exceedingly  burdensome.  The  Latin  equivalents 
for  his  Greek  terms  are  found  in  the  printed  arithmetics  of  his 

disciples,  1500  years  later.     Thus  the  ratio  m  "*"      is  super- 

wi 

particularis,  — -  - —  is   subsuperparticularis,   3J  = — —  is 

inn  -f- 1  4 

triplex  sesquiquartus.1  Nicomachus  gives  tables  of  numbers  in 
form  of  a  chess-board  of  100  squares.  It  might  have  answered 
as  a  multiplication-table,  but  it  appears  to  have  been  used  in 
the  study  of  ratios.2  He  describes  polygonal  numbers,  the 
different  kinds  of  proportions  (11  in  all),  and  treats  of  the 
summation  of  numerical  series.  To  be  noted  is  the  absence  of 
rules  of  computation,  of  problem-working,  and  of  practical 
arithmetic.  He  gives  the  following  important  proposition. 
All  cubical  numbers  are  equal  to  the  sum  of  successive  odd 

1  Gow,  pp.  90,  91. 

2FRiEDLBiN,  p.  78 ;  CANTOR,  Vol.  I.,  p.  481. 


GREECE  33 

numbers.     Thus,  8  =  23  =  3  +  5  ;   27  =  33  =  7  +  9  +  11;    64  = 
43  =  13  +  15  +  17  +  19. 

In  the  writings  of  Nicomachus,  lamblichus,  Theon  of 
Smyrna,  Thymaridas,  and  others,  are  found  investigations 
algebraic  in  their  nature.  Thymaridas  in  one  place  uses  a 
Greek  word  meaning  "  unknown  quantity"  in  a  manner  sug- 
gesting the  near  approach  of  algebra.  Of  interest  in  tracing 
the  evolution  of  algebra  are  the  arithmetical  epigrams  in  the 
Palatine  Anthology,  which  contained  about  50  problems  leading 
to  linear  equations.1  Before  the  introduction  of  algebra,  these 
problems  were  propounded  as  puzzles.  No.  23  gives  the  times 
in  which  four  fountains  can  fill  a  reservoir  separately  and 
requires  the  time  they  can  fill  it  conjointly.2  No.  9.  What 
part  of  the  day  has  disappeared,  if  the  time  left  is  twice  two- 
thirds  of  the  time  passed  away  ?  Sometimes  included  among 
these  epigrams  is  the  famous  "  cattle-problem,"  which  Archi- 
medes is  said  to  have  propounded  to  the  Alexandrian  mathe- 
maticians.3 This  difficult  problem  is  indeterminate.  In  the 
first  part  of  it,  from  only  seven  equations,  eight  unknown 
quantities  in  integral  numbers  are  to  be  found.  Gow  states 
it  thus:  The  sun  had  a  herd  of  bulls  and  cows,  of  different 
colours.  (1)  Of  Bulls,  the  white  (  W)  were  in  number  (i  +  J) 
of  the  blue  (B)  and  yellow  (F)  ;  the  B  were  (\  +  £)  of  the  Y 
and  piebald  (P)  ;  the  P  were  (1  +  -f  )  of  the  W  and  F.  (2)  Of 
Cows,  which  had  the  same  colours  (w,  b,  y,p),  w  =  (1  +  %)(B  +  6); 


1  These  epigrams  were  written  in  Greek,  perhaps  about  the  time  of 
Constantine  the  Great.    For  a  German  translation,  see  G.   WERTHEIM, 
Die  Arithmetik  und  die  Schrift  itber  Polygonalzahlen  des  Diophantus  von 
Alexandria,  Leipzig,  1890,  pp.  330-344. 

2  WERTHEIM,  op.  cit.,  p.  337. 

8  Whether  it  originated  at  the  time  of  Archimedes  or  later  is  discussed 
by  T.  L.  HEATH,  Diophantos  of  Alexandria,  Cambridge,  1885,  pp.  142- 
147. 


34  A  HISTORY  OF   MATHEMATICS 

Find  the  number  of  bulls  and  cows.  This  leads  to  excessively 
high  numbers,  but  to  add  to  its  complexity,  a  second  series  of 
conditions  is  superadded,  leading  to  an  indeterminate  equa- 
tion of  the  second  degree. 

Most  of  the  problems  in  the  Palatine  Anthology,  though 
puzzling  to  an  arithmetician,  are  easy  to  an  algebraist.  Such 
problems  became  popular  about  the  time  of  Diophantus  and 
doubtless  acted  as  a  powerful  mental  stimulus. 

Diophantus,  one  of  the  last  Alexandrian  mathematicians,  is 
generally  regarded  as  an  algebraist  of  great  fertility.1  He 
died  about  330  A.D.  His  age  was  84,  as  is  known  from  an 
epitaph  to  the  following  effect:  Diophantus  passed  £  of  his 
life  in  childhood,  y1^  in  youth,  and  \  more  as  a  bachelor ;  five 
years  after  his  marriage,  was  born  a  son  who  died  four  years 
before  his  father,  at  half  his  father's  age.  This  epitaph  states 
about  all  we  know  of  Diophantus.  We  are  uncertain  as  to  the 
time  of  his  death  and  ignorant  of  his  parentage  and  place  of 
nativity.  Were  his  works  not  written  in  Greek,  no  one  would 
suspect  them  of  being  the  product  of  Greek  mind.  The  spirit 
pervading  his  masterpiece,  the  Arithmetica  [said  to  have  been 
written  in  thirteen  books,  of  which  only  six  (seven  ?)2  are  ex- 
tant] is  as  different  from  that  of  the  great  classical  works  of 
the  time  of  Euclid  as  pure  geometry  is  from  pure  analysis. 
Among  the  Greeks,  Diophantus  had  no  prominent  forerunner, 
no  prominent  disciple.  Except  for  his  works,  we  should  be 
obliged  to  say  that  the  Greek  mind  accomplished  nothing 
notable  in  algebra.  Before  the  discovery  of  the  Ahmes 
papyrus,  the  Arithmetica  of  Diophantus  was  the  oldest  known 
work  on  algebra.  Diophantus  introduces  the  notion  of  an 
algebraic  equation  expressed  in  symbols.  Being  completely 

1  "How  far  was  Diophantos  original?"  see  HEATH,  op.  cit.,  pp.  133- 
159. 

2  CANTOR,  Vol.  I.,  pp.  466,  467. 


GREECE  35 

divorced  from  geometry,  his  treatment  is  purely  analytical. 
He  is  the  first  to  say  that  "  a  number  to  be  subtracted  multi- 
plied by  a  number  to  be  subtracted  gives  a  number  to  be 
added."  This  is  applied  to  differences,  like  (2  x  -  3)  (2  x  -  3), 
the  product  of  which  he  finds  without  resorting  to  geometry. 
Identities  like  (a  -f  6)2  =  a2  +  2  ab  +  62,  which  are  elevated 
by  Euclid  to  the  exalted  rank  of  geometric  theorems,  with 
Diophantus  are  the  simplest  consequences  of  algebraic  laws  of 
operation.  Diophantus  represents  the  unknown  quantity  x  by 
s',  the  square  of  the  unknown  x2  by  8",  x3  by  K",  x4  by  88V. 
His  sign  for  subtraction  is  >7>,  his  symbol  for  equality,  i.  Addi- 
tion is  indicated  by  juxtaposition.  Sometimes  he  ignores  these 
symbols  and  describes  operations  in  words,  when  the  symbols 
would  have  answered  better.  In  a  polynomial  all  the  positive 
terms  are  written  before  any  of  the  negative  ones.  Thus, 
x3  —  5  ce2  +  8 x  —  1  would  be  in  his  notation,1  K5ds°1^  /p^e/d. 
Here  the  numerical  coefficient  follows  the  x. 

To  be  emphasized  is  the  fact  that  in  Diophantus  the  funda- 
mental algebraic  conception  of  negative  numbers  is  wanting. 
In  2  x  — 10  he  avoids  as  absurd  all  cases  where  2  x  <  10. 
Take  Probl.  16  Bk.  I.  in  his  Arithmetica:  "To  find  three  num- 
bers such  that  the  sums  of  each  pair  are  given  numbers."  If 
a,  b)  c  are  the  given  numbers,  then  one  of  the  required  numbers 
is  \(a  -f  b  -f-  c)  —  c.  If  c  >  ^(a  -f-  b  +  c),  then  this  result  is  unin- 
telligible to  Diophantus.  Hence  he  imposes  upon  the  problem 
the  limitation,  "  But  half  the  sum  of  the  three  given  numbers 
must  be  greater  than  any  one  singly."  Diophantus  does  not 
give  solutions  general  in  form.  In  the  present  instance  the 
special  values  20,  30,  40  are  assumed  as  the  given  numbers. 

In  problems  leading  to  simultaneous  equations  Diophantus 
adroitly  uses  only  one  symbol  for  the  unknown  quantities. 

1  HEATH,  op.  cit.,  p.  72. 


36  A    HISTORY   OF   MATHEMATICS 

This  poverty  in  notation  is  offset  in  many  cases  merely 
by  skill  in  the  selection  of  the  unknown.  Frequently  he 
follows  a  method  resembling  somewhat  the  Hindu  "false 
position  " :  a  preliminary  value  is  assigned  to  the  unknown 
which  satisfies  only  one  or  two  of  the  necessary  conditions. 
This  leads  to  expressions  palpably  wrong,  but  nevertheless 
suggesting  some  stratagem  by  which  one  of  the  correct 
values  can  be  obtained.1 

Diophantus  knows  how  to  solve  quadratic  equations,  but  in 
the  extant  books  of  his  Arithmetica  he  nowhere  explains  the 
mode  of  solution.  Noteworthy  is  the  fact  that  he  always  gives 
but  one  of  the  two  roots,  even  when  both  roots  are  positive. 
Nor  does  he  ever  accept  as  an  answer  to  a  problem  a  quantity 
which  is  negative  or  irrational. 

Only  the  first  book  in  the  Arithmetica  is  devoted  to  deter- 
minate equations.  It  is  in  the  solution  of  indeterminate 
equations  (of  the  second  degree)  that  he  exhibits  his  wonderful 
inventive  faculties.  However,  his  extraordinary  ability  lies 
less  in  discovering  general  methods  than  in  reducing  all  sorts 
of  equations  to  particular  forms  which  he  knows  how  to  solve. 
Each  of  his  numerous  and  various  problems  has  its  own  distinct 
method  of  solution,  which  is  often  useless  in  the  most  closely 
related  problem.  "It  is,  therefore,  difficult  for  a  modern 
mathematician,  after  studying  100  Diophantine  solutions, 
to  solve  the  101st.  .  .  .  Diophantus  dazzles  more  than  he 
delights."2 

The  absence  in  Diophantus  of  general  methods  for  dealing 
with  indeterminate  problems  compelled  modern  workers  on 
this  subject,  such  as  Euler,  Lagrange,  Gauss,  to  begin  anew. 

1  Gow,  pp.  110,  116,  117. 

2  HANKEL,  p.  165.     It  should  be  remarked  that  HEATH,  op.  cit.,  pp. 
83-120,  takes  exception   to  HankePs  verdict  and  endeavours  to  give  a 
general  account  of  Diophantine  methods. 


ROME  37 

Diophantus  could  teach  them  nothing  in  the  way  of  general 
methods.  The  result  is  that  the  modern  theory  of  numbers  is 
quite  distinct  and  a  decidedly  higher  and  nobler  science  than 
Diophantine  Analysis.  Modern  disciples  of  Diophantus  usually 
display  the  weaknesses  of  their  master  and  for  that  reason 
have  failed  to  make  substantial  contributions  to  the  subject. 

Of  special  interest  to  us  is  the  method  followed  by  Diophan- 
tus in  solving  a  linear  determinate  equation.  His  directions 
are :  "  If  now  in  any  problem  the  same  powers  of  the  unknown 
occur  on  both  sides  of  the  equation,  but  with  different  coeffi- 
cients, we  must  subtract  equals  from  equals  until  we  have  one 
term  equal  to  one  term.  If  there  are  on  one  side,  or  on  both 
sides,  terms  with  negative  coefficients,  these  terms  must  be 
added  on  both  sides,  so  that  on  both  sides  there  are  only  posi- 
tive terms.  Then  we  must  again  subtract  equals  from  equals 
until  there  remains  only  one  term  in  each  number."1  Thus 
what  is  nowadays  achieved  by  transposing,  simplifying,  and 
dividing  by  the  coefficient  of  x9  was  accomplished  by  Diophan- 
tus by  addition  and  subtraction.  It  is  to  be  observed  that  in 
Diophantus,  and  in  fact  in  all  writings  of  antiquity,  the  con- 
ception of  a  quotient  is  wanting.  An  operation  of  division  is 
nowhere  exhibited.  When  one  number  had  to  be  divided  by 
another,  the  answer  was  reached  by  repeated  subtractions.2 

KOME 

Of  Roman  methods  of  computation  more  is  known  than  of 
Greek  or  Egyptian.  Abacal  reckoning  was  taught  in  schools. 
Writers  refer  to  pebbles  and  a  dust-covered  abacus,  ruled 
into  columns.  An  Etruscan  (?)  relic,  now  preserved  in  Paris, 
shows  a  computer  holding  in  his  left  hand  an  abacus  with 

1  WERTHEIM'S  Diophantus,  p.  7. 

2  NESSELMANN,  p.  112  ;  FRIEDLEIN,  p.  79. 


38 


A   HISTORY  OF   MATHEMATICS 


numerals  set  in  columns,  while  with  the  right  hand  he  lays 
pebbles  upon  the  table.1 

The  Romans  used  also  another  kind  of  abacus,  consisting  of 
a  metallic  plate  having  grooves  with  movable  buttons.  By  its 
use  all  integers  between  1  and  9,999,999,  as  well  as  some  frac- 
tions, could  be  represented.  In  the  two  adjoining  figures 
(taken  from  Fig.  21  in  Friedlein)  the  lines  represent  grooves 


IF  C  x  i  c  x  i 


Ilium 

•XI          IT  c  x  i  c  x  i  -jC 


i 


and  the  circles  buttons.  The  Roman  numerals  indicate  the 
value  of  each  button  in  the  corresponding  groove  below, 
the  button  in  the  shorter  groove  above  having  a  fivefold  value. 
Thus  II  =  1,000,000 ;  hence  each  button  in  the  long  left-hand 
groove,  when  in  use,  stands  for  1,000,000,  and  the  button  in  the 
short  upper  groove  stands  for  5,000,000.  The  same  holds  for 
the  other  grooves  labelled  by  Roman  numerals.  The  eighth 
long  groove  from  the  left  (having  5  buttons)  represents  duo- 
decimal fractions,  each  button  indicating  y1^,  while  the  button 
above  the  dot  means  T%.  In  the  ninth  column  the  upper 
button  represents  ^T,  the  middle  -fa,  and  two  lower  each  TL. 
Our  first  figure  represents  the  positions  of  the  buttons  before 
the  operation  begins  ;  our  second  figure  stands  for  the  number 
852  £J¥.  The  eye  has  here  to  distinguish  the  buttons  in  use 
and  those  left  idle.  Those  counted  are  one  button  above 


1  CANTOR,  Vol.  I.,  p.  529. 


ROME  39 

c  (=500),  and  three  buttons  below  c(=300);  one  button 
above  x  (=  50)  ;  two  buttons  below  I  (=  2)  ;  four  buttons 
indicating  duodecimals  (=  -J.)  j  and  the  button  for  ^V 

Suppose  now  that  10,318  £  -J-  ^  is  to  be  added  to  852  -J-  ^. 
The  operator  could  begin  with  the  highest  units,  or  the  lowest 
units,  as  he  pleased.  Naturally  the  hardest  part  is  the  addi- 
tion of  the  fractions.  In  this  case  the  button  for  J^,  the  button 
above  the  dot  and  three  buttons  below  the  dot  were  used  to 
indicate  the  sum  f  -fa.  The  addition  of  8  would  bring  all  the 
buttons  above  and  below  1  into  play,  making  10  units.  Hence, 
move  them  all  back  and  move  up  one  button  in  the  groove 
below  x.  Add  10  by  moving  up  another  of  the  buttons  below 
x  ;  add  300  to  800  by  moving  back  all  buttons  above  and  below 
c,  except  one  button  below,  and  moving  up  one  button  below  I  ; 
add  10,000  by  moving  up  one  button  below  x.  In  subtraction 
the  operation  was  similar. 

Multiplication  could  be  carried  out  in  several  ways.  In  case 
of  38  %  2T  times  25  J,  the  abacus  may  have  shown  successively 
the  following  values  :  600  (=  30  •  20),  760  (=  600  +  20  •  8), 
770  (=  760  +  1  -  20),  770  |f  (=  770  +  ^  .  20),  920  if  (=  770|f 
+  30  •  5),  960if  (=  920  |f  +  8-5),  9631  (=  960  if  +  £  .  5), 
9631  A  (=963^  +  A  •  5),  973  |  ^  (=  963  i  ^  +  i  '  30)> 


In  division  the  abacus  was  used  to  represent  the  remainder 
resulting  from  the  subtraction  from  the  dividend  of  the  divisor 
or  of  a  convenient  multiple  of  the  divisor.  The  process  was 
complicated  and  difficult.  These  methods  of  abacal  computa- 
tion show  clearly  how  multiplication  or  division  can  be 
carried  out  by  a  series  of  successive  additions  or  subtractions. 
In  this  connection  we  suspect  that  recourse  was  had  to  mental 

1  ERIEDLEIX,  p.  89. 


40  A    HISTORY   OF   MATHEMATICS 

operations  and  to  the  multiplication  table.  Finger-reckoning 
may  also  have  been  used.  In  any  case  the  multiplication  and 
division  with  large  numbers  must  have  been  beyond  the  power 
of  the  ordinary  computer.  This  difficulty  was  sometimes 
obviated  by  the  use  of  arithmetical  tables  from  which  the 
required  sum,  difference,  or  product  of  two  numbers  could  be 
copied.  Tables  of  this  sort  were  prepared  by  Victorius  of 
Aquitania,  a  writer  who  is  well  known  for  his  canon  paschalis, 
a  rule  for  rinding  the  correct  date  for  Easter,  which  he  pub- 
lished in  457  A..D.  The  tables  of  Victorius  contain  a  peculiar 
notation  for  fractions,  which  continued  in  use  throughout  the 
middle  ages.1  Fractions  occur  among  the  Romans  most  fre- 
quently in  money  computations. 

The  Eoman  partiality  to  duodecimal  fractions  is  to  be 
observed.  Why  duodecimals  and  not  decimals  ?  Doubtless 
because  the  decimal  division  of  weights  and  measures  seemed 
unnatural.  In  everyday  affairs  the  division  of  units  into  2,  3, 
4,  6  equal  parts  is  the  commonest,  and  duodecimal  fractions 
give  easier  expressions  for  these  parts.  In  duodecimals  the 
above  parts  are  •£%,  -^,  -f^  -^  of  the  whole ;  in  decimals  these 

5    3—  2—  1— 
parts  are  ^  -^-,  ^  ^.     Unlike  the  Greeks,  the  Romans  dealt 

with  concrete  fractions.  The  Roman  as,  originally  a  copper 
coin  weighing  one  pound,  was  divided  into  12  uncice.  The 
abstract  fraction  J-i  was  expressed  concretely  by  deunx  (==  de 
uncia,  i.e.,  as  [1]  less  uncia  [y1^])  >  A  was  ca^e(i  quincunx 
(=  quinque  [five]  uncice) ;  thus  each  Roman  fraction  had  a 
special  name.  Addition  and  subtraction  of  such  fractions  were 
easy.  Fractional  computations  were  the  chief  part  of  arith- 
metical instruction  in  Roman  schools.2  Horace,  in  remem- 
brance, perhaps,  of  his  own  school-days,  gives  the  following 

1  Consult  FRIEDLEIN,  pp.  93-98.  2  HANKEL,  pp.  58,  59. 


KOME  41 

dialogue  between  teacher  and  pupil  ( ATS  poetica,  V.  326-330)  : 
"  Let  the  son  of  Albinus  tell  me,  if  from  five  ounces  [i.e.  -j^-] 
be  subtracted  one  ounce  [i.e.  Ty ,  what  is  the  remainder  ? 
Come,  you  can  tell.  'One-third.'  Good;  you  will  be  able  to 
take  care  of  your  property.  If  one  ounce  [i.e.  y1^-]  be  added, 
what  does  it  make  ?  ' One-half.' " 

Doubtless  the  Romans  unconsciously  hit  upon  a  fine 
pedagogical  idea  in  their  concrete  treatment  of  fractions. 
Roman  boys  learned  fractions  in  connection  with  money, 
weights,  and  measures.  We  conjecture  that  to  them  fractions 
meant  more  than  what  was  conveyed  by  the  definition, 
"  broken  number,"  given  in  old  English  arithmetics. 

One  of  the  last  Roman  writers  was  Boethius  (died  524),  who 
is  to  be  mentioned  in  this  connection  as  the  author  of  a  work, 
De  Institutions  Arithmetics,  essentially  a  translation  of  the 
arithmetic  of  Nicomachus,  although  some  of  the  most  beau- 
tiful arithmetical  results  in  the  original  are  omitted  by 
Boethius.  The  historical  importance  of  this  translation  lies 
in  the  extended  use  made  of  it  later  in  Western  Europe. 

The  Roman  laws  of  inheritance  gave  rise  to  numerous  arith- 
metical examples.  The  following  is  of  interest  in  itself,  and 
also  because  its  occurrence  elsewhere  at  a  later  period  assists 
us  in  tracing  the  source  of  arithmetical  knowledge  in  Western 
Europe :  A  dying  man  wills,  that  if  his  wife,  being  with  child, 
gives  birth  to  a  son,  the  son  shall  receive  -§- ,  and  she  \  of  his 
estate  ;  but  if  a  daughter  is  born,  the  daughter  shall  receive 
^,  and  the  wife  \.  It  happens  that  twins  are  born,  a  boy  and 
a  girl.  How  shall  the  estate  be  divided  so  as  to  satisfy  the 
will?  The  celebrated  Roman  jurist,  Salvianus  Julianus, 
decided  that  it  should  be  divided  into  seven  equal  parts,  of 
which  four  should  go  to  the  son,  two  to  the  wife  and  one 
to  the  daughter. 

Aside  from  the  (probable)  improvement  of  the  abacus  and 


A   HISTORY   OF   MATHEMATICS 

the  development  of  duodecimal  fractions,  the  Komans  made  no 
contributions  to  arithmetic.  The  algebra  of  Diophantus  was 
unknown  to  them.  With  them  as  with  all  nations  of  antiquity 
numerical  calculations  were  long  and  tedious,  for  the  reason 
that  they  never  possessed  the  boon  of  a  perfect  notation  of 
numbers  with  its  zero  and  principle  of  local  value. 


GEOMETRY  AND   TRIGONOMETRY 


EGYPT  AND   BABYLONIA 

THE  crude  beginnings  of  empirical  geometry,  like  the  art  of 
counting,  must  be  of  very  ancient  origin.  We  suspect  that  our 
earliest  records,  reaching  back  to  about  2500  B.C.,  represent 
comparatively  modern  thought.  The  Ahmes  papyrus  and  the 
Egyptian  pyramids  are  probably  the  oldest  evidences  of  geo- 
metrical study.  We  find  it  more  convenient,  however,  to 
begin  with  Babylonia.  Ancient  science  is  closely  knitted  with 
superstition.  We  have  proofs  that  in  Babylonia  geometrical 
figures  were  used  in  augury.1  Among  these  figures  are  a  pair 
of  parallel  lines,  a  square,  a  figure  with  a  re-entrant  angle,  and 
an  incomplete  figure,  believed  to  represent  three  concentric  tri- 
angles with  their  sides  respectively  parallel.  The  accompany- 
ing text  contains  the  Sumerian  word  tim,  meaning  "line," 
originally  "rope";  hence  the  conjecture  that  the  Babylonians, 
like  the  Egyptians,  used  ropes  in  measuring  distances,  and  in 
determining  certain  angles.  The  Babylonian  sign  •£  is  be- 
lieved to  be  associated  with  the  division  of  the  circle  into  six 
equal  parts,  and  (as  the  Babylonians  divided  the  circle  into 
360  degrees)  with  the  origin  of  the  sexagesimal  system.  That 
this  division  into  six  parts  (probably  by  the  sixfold  application 
of  the  radius)  was  known  in  Babylonia,  follows  from  the  in- 
spection of  the  six  spokes  in  the  wheel  of  a  royal  carriage 

1  CANTOR,  Vol.  L,  pp.  45-48. 
43 


44  A  HISTORY   OF   MATHEMATICS 

represented  in  a  drawing  found  in  the  remains  of  Nineveh. 
Like  the  Hebrews  (1  Kings  vii.  23),  the  Babylonians  took  the 
ratio  of  the  circumference  to  the  diameter  equal  to  3,  a  de- 
cidedly inaccurate  value.  Of  geometrical  demonstrations  there 
is  no  trace.  "As  a  rule,  in  the  Oriental  mind  the  intuitive 
powers  eclipse  the  severely  rational  and  logical." 

We  begin  our  account  of  Egyptian  geometry  with  the  geo- 
metrical problems  of  the  Ahmes  papyrus,  which  are  found  in 
the  middle  of  the  arithmetical  matter.  Calculations  of  the 
solid  contents  of  barns  precede  the  determination  of  areas.1 
Not  knowing  the  shape  of  the  barns,  we  cannot  verify  the  cor- 
rectness of  the  computations,  but  in  plane  geometry  Ahmes's 
figures  usually  help  us.  He  considers  the  area  of  land  in  the 
forms  of  square,  oblong,  isosceles  triangle,  isosceles  trapezoid, 
and  circle.  Example  No.  44  gives  100  as  the  area  of  a  square 
whose  sides  are  10.  In  No.  51  he  draws  an  isosceles  triangle 
whose  sides  are  10  ruths  and  whose  base  is  4  ruths,  and  finds 
the  area  to  be  20.  The  correct  value  is  19.6".  Ahmes's 
approximation  is  obtained  by  taking  the  product  of  one  leg 
and  half  the  base.  The  same  error  occurs  in  the  area  of 
the  isosceles  trapezoid.  Half  the  sum  of  the  two  bases  is 
multiplied  by  one  leg.  His  treatment  of  the  circle  is  an  actual 
quadrature,  for  it  teaches  how  to  find  a  square  equivalent  to 
the  circular  area.  He  takes  as  the  side,  the  diameter  dimin- 
ished by  -^  of  itself.  This  is  a  fair  approximation,  for,  if  the 
radius  is  taken  as  unity,  then  the  side  of  the  square  is  ^-,  and 
its  area  (^)2  =  3.1604  ...  Besides  these  problems  there  are 
others  relating  to  pyramids  and  disclosing  some  knowledge  of 
similar  figures,  of  proportion,  and,  perhaps,  of  rudimentary 
trigonometry.2 

Besides   the   Ahmes   papyrus,    proofs   of  the   existence   of 

1  Gow,  pp.  126-130. 

2  Consult  CANTOR,  Vol.  I.,  pp.  99-102  ;  Gow,  p.  128. 


EGYPT   AND   BABYLONIA  45 

ancient  Egyptian  geometry  are  found  in  figures  on  the  walls 
of  old  structures.  The  wall  was  ruled  with  squares,  or  other 
rectilinear  figures,  within  which  coloured  pictures  were  drawn.1 

The  Greek  philosopher  Democritus  (about  460-370  B.C.)  is 
quoted  as  saying  that  "  in  the  construction  of  plane  figures  .  .  . 
no  one  has  yet  surpassed  me,  not  even  the  so-called  Harpedo- 
naptce  of  Egypt."  Cantor  has  pointed  out  the  meaning  of  the 
word  "  harpedonaptse  "  to  be  "  rope-stretchers." 2  This,  together 
with  other  clues,  led  him  to  the  conclusion  that  in  laying  out 
temples  the  Egyptians  determined  by  accurate  astronomical 
observation  a  north  and  south  line ;  then  they  constructed  a 
line  at  right  angles  to  this  by  means  of  a  rope  stretched  around 
three  pegs  in  such  a  way  that  the  three  sides  of  the  triangle 
formed  are  to  each  other  as  3 : 4 : 5,  and  that  one  of  the  legs  of 
this  right  triangle  coincided  with  the  N.  and  S.  line.  Then 
the  other  leg  gave  the  E.  and  W.  line  for  the  exact  orientation 
of  the  temple.  According  to  a  leathern  document  in  the  Ber- 
lin Museum,  "  rope-stretching "  occurred  at  the  very  early  age 
of  Amenemhat  I.  If  Cantor's  explanation  is  correct,  it  fol- 
lows, therefore,  that  the  Egyptians  were  familiar  with  the 
well-known  property  of  the  right-triangle,  in  case  of  sides  in 
the  ratio  3:4:5,  as  early  as  2000  B.C. 

From  what  has  been  said  it  follows  that  Egyptian  geometry 
flourished  at  a  very  early  age.  What  about  its  progress  in 
subsequent  centuries  ?  In  257  B.C.  was  laid  out  the  temple  of 
Horus,  at  Edfu,  in  Upper  Egypt.  About  100  B.C.  the  number 
of  pieces  of  land  owned  by  the  priesthood,  and  their  areas, 
were  inscribed  upon  the  walls  in  hieroglyphics.  The  incorrect 
formula  of  Ahmes  for  the  isosceles  trapezoid  is  here  applied 
for  any  trapezium,  however  irregular.  Thus  the  formulae  of 
more  than  2000  B.C.  yield  closer  approximations  than  those 

1  Consult  the  drawings  reproduced  in  CANTOR,  Vol.  I. ,  p.  108. 

2  Ibidem,  p.  104. 


46  A   HISTORY   OF  MATHEMATICS 

written  two  centuries  after  Euclid!  The  conclusion  irresist- 
ibly follows  that  the  Egyptians  resembled  the  Chinese  in  the 
stationary  character,  not  only  of  their  government,  but  also  of 
their  science.  An  explanation  of  this  has  been  sought  in  the 
fact  that  their  discoveries  in  mathematics,  as  also  in  medicine, 
were  entered  at  an  early  time  upon  their  sacred  books,  and 
that,  in  after  ages,  it  was  considered  heretical  to  modify  or 
augment  anything  therein.  Thus  the  books  themselves  closed 
the  gates  to  progress. 

Egyptian  geometry  is  mainly,  though  not  entirely,  a  geome- 
try of  areas,  for  the  measurement  of  figures  and  of  solids  con- 
stitutes the  main  part  of  it.  This  practical  geometry  can 
hardly  be  called  a  science.  In  vain  we  look  for  theorems  and 
proofs,  or  for  a  logical  system  based  on  axioms  and  postulates. 
It  is  evident  that  some  of  the  rules  were  purely  empirical. 

If  we  may  trust  the  testimony  of  the  Greeks,  Egyptiar 
geometry  had  its  origin  in  the  surveying  of  land  necessitated, 
by  the  frequent  overflow  of  the  Nile. 


GEEECE 

About  the  seventh  century  B.C.  there  arose  between  Greece 
and  Egypt  a  lively  intellectual  as  well  as  commercial  inter- 
course. Just  as  Americans  in  our  time  go  to  Germany  to 
study,  so  early  Greek  scholars  visited  the  land  of  the  pyra- 
mids. Thales,  CEnopides,  Pythagoras,  Plato,  Democritus, 
Eudoxus,  all  sat  at  the  feet  of  the  Egyptian  priests  for  in- 
struction. While  Greek  culture  is,  therefore,  not  primitive, 
it  commands  our  enthusiastic  admiration.  The  speculative 
mind  of  the  Greek  at  once  transcended  questions  pertaining 
merely  to  the  practical  wants  of  everyday  life;  it  pierced 
into  the  ideal  relations  of  things,  and  revelled  in  the  study 


GREECE  47 

of  science  as  science.  For  this  reason  Greek  geometry  will 
always  be  admired,  notwithstanding  its  limitations  and 
defects. 

Eudemus,  a  pupil  of  Aristotle,  wrote  a  history  of  geometry. 
This  history  has  been  lost ;  but  an  abstract  of  it,  made  by  Pro- 
clus  in  his  commentaries  on  Euclid,  is  extant,  and  is  the  most 
trustworthy  information  we  have  regarding  early  Greek 
geometry.  We  shall  quote  the  account  under  the  name  of 
Eudemian  Summary. 

I.  The  Ionic  School.  —  The  study  of  geometry  was  introduced 
into  Greece  by  Tholes  of  Miletus  (640-546  B.C.),  one  of  the 
"seven  wise  men."  Commercial  pursuits  brought  him  to 
Egypt;  intellectual  pursuits,  for  a  time,  detained  him  there. 
Plutarch  declares  that  Thales  soon  excelled  the  priests,  and 
amazed  King  Amasis  by  measuring  the  heights  of  the  pyra- 
mids from  their  shadows.  According  to  Plutarch  this  was 
done  thus :  The  length  of  the  shadow  of  the  pyramid  is  to  the 
shadow  of  a  vertical  staff,  as  the  unknown  height  of  the  pyra- 
mid is  to  the  known  length  of  the  staff.  But  according  to 
Diogenes  Laertius  the  measurement  was  different :  The  height 
of  the  pyramid  was  taken  equal  to  the  length  of  its  shadow  at 
the  moment  when  the  shadow  of  a  vertical  staff  was  equal  to 
its  own  length.1 

1  The  first  method  implies  a  knowledge  of  the  proportionality  of  the 
sides  of  equiangular  triangles,  which  some  critics  are  unwilling  to  grant 
Thales.  The  rudiments  of  proportion  were  certainly  known  to  Ahmes 
and  the  builders  of  the  pyramids.  Allman  grants  Thales  this  knowledge, 
and,  in  general,  assigns  to  him  and  his  school  a  high  rank.  See  Greek 
Geometry  from  Thales  to  Euclid,  by  GEORGE  JOHNSTON  ALLMAN,  Dublin, 
1889,  p.  14.  Gow  (p.  142)  also  believes  in  Plutarch's  narrative,  but 
Cantor  (I.,  p.  135)  leaves  the  question  open,  while  Hankel  (p.  90),  Bret- 
schneider,  Tannery,  Loria,  are  inclined  to  deny  Thales  a  knowledge  of 
similitude  of  figures.  See  Die  Geometric  und  die  Geometer  vor  Euklides. 
Ein  Historischer  Versuch,  von  C.  A.  BRETSCHNEIDER,  Leipzig,  1870,  p.  46  ; 
La  Geometrie  Grecque  ...  par  PAUL  TANNERY,  Paris,  1887,  p.  92  ;  Le 


48  A   HISTORY   OF   MATHEMATICS 

The  Eudemian  Summary  ascribes  to  Thales  the  invention  of 
the  theorems  on  the  equality  of  vertical  angles,  the  equality 
of  the  base  angles  in  isosceles  triangles,  the  bisection  of  the 
circle  by  any  diameter,  and  the  congruence  of  two  triangles 
having  a  side  and  the  two  adjacent  angles  equal  respectively. 
Famous  is  his  application  of  the  last  theorem  to  the  deter- 
mination of  the  distances  of  ships  from  the  shore.  The 
theorem  that  all  angles  inscribed  in  a  semicircle  are  right 
angles  is  attributed  by  some  ancients  to  Thales,  by  others  to 
Pythagoras.  Thales  thus  seems  to  have  originated  the  geome- 
try of  lines  and  of  angles,  essentially  abstract  in  character, 
while  the  Egyptians  dealt  primarily  with  the  geometry  of  sur- 
faces and  of  solids,  empirical  in  character.1  It  would  seem  as 
though  the  Egyptian  priests  who  cultivated  geometry  ought  at 
least  to  have  felt  the  truth  of  the  above  theorems.  We  have 
no  doubt  that  they  did,  but  we  incline  to  the  opinion  that 
Thales,  like  a  true  philosopher,  formulated  into  theorems  and 
subjected  to  proof  that  which  others  merely  felt  to  be  true. 
If  this  view  is  correct,  then  it  follows  that  Thales  in  his  pyra- 
mid and  ship  measurements  was  the  first  to  apply  theoretical 
geometry  to  practical  uses. 

Thales  acquired  great  celebrity  by  the  prediction  of  a  solar 
eclipse  in  585  B.C.  With  him  begins  the  study  of  scientific 
astronomy.  The  story  goes  that  once,  while  viewing  the  stars, 
he  fell  into  a  ditch.  An  old  woman  attending  him  exclaimed, 
"  How  canst  thou  know  what  is  doing  in  the  heavens,  when 
thou  seest  not  what  is  at  thy  feet  ?  " 

Astronomers  of  the  Ionic  school  are  Anaximander  and 
Anaximenes.  Anaxagoras,  a  pupil  of  the  latter,  attempted  to 
square  the  circle  while  he  was  confined  in  prison.  Approxi- 

Scienze  Esatte  nelV  Antica  Grecia,  di  GINO  LORIA,  in  Modena,  1893, 
Libro  L,  p.  25. 
1  ALLMAN,  p.  16. 


GREECE  49 

mations  to  the  ratio  TT  occur  early  among  the  Egyptians,  Baby- 
lonians, and  Hebrews ;  but  Anaxagoras  is  the  first  recorded  as 
attempting  the  determination  of  the  exact  ratio  —  that  knotty 
problem  which  since  his  time  has  been  unsuccessfully  attacked 
by  thousands.  Anaxagoras  apparently  offered  no  solution. 

II.  The  Pythagorean  School  —  The  life  of  Pythagoras 
(580?-500?  B.C.)  is  enveloped  in  a  deep  mythical  haze.  We 
are  reasonably  certain,  however,  that  he  was  born  in  Samos, 
studied  in  Egypt,  and,  later,  returned  to  the  place  of  his 
nativity.  Perhaps  he  visited  Babylon.  Failing  in  his  attempt 
to  found  a  school  in  Samos,  he  followed  the  current  of  civiliza- 
tion, and  settled  at  Croton  in  South  Italy  (Magna  Graecia). 
There  he  founded  the  Pythagorean  brotherhood  with  observ- 
ances approaching  Masonic  peculiarity.  Its  members  were 
forbidden  to  divulge  the  discoveries  and  doctrines  of  their 
school.  Hence  it  is  now  impossible  to  tell  to  whom  individu- 
ally the  various  Pythagorean  discoveries  must  be  ascribed.  It 
was  the  custom  among  the  Pythagoreans  to  refer  every  dis- 
covery to  the  great  founder  of  the  sect.  At  first  the  school 
flourished,  but  later  it  became  an  object  of  suspicion  on  account 
of  its  mystic  observances.  A  political  party  in  Lower  Italy 
destroyed  the  buildings ;  Pythagoras  fled,  but  was  killed  at 
Metapontum.  Though  politics  broke  up  the  Pythagorean  fra- 
ternity, the  school  continued  to  exist  for  at  least  two  centuries. 

Like  Thales,  Pythagoras  wrote  no  mathematical  treatises. 
The  Eudemian  Summary  says :  "  Pythagoras  changed  the  study 
of  geometry  into  the  form  of  a  liberal  education,  for  he  exam- 
ined its  principles  to  the  bottom,  and  investigated  its  theorems 
in  an  immaterial  and  intellectual  manner." 

To  Pythagoras  himself  is  to  be  ascribed  the  well-known 
property  of  the  right  triangle.  The  truth  of  the  theorem  for 
the  special  case  when  the  sides  are  3,  4,  and  5,  respectively,  he 
may  have  learned  from  the  Egyptians.  We  are  told  that  Py- 


50  A    HISTORY   OF   MATHEMATICS 

thagoras  was  so  jubilant  over  this  great  discovery  that  he  sac- 
rificed a  hecatomb  to  the  muses  who  inspired  him.  That  this 
is  but  a  legend  seems  plain  from  the  fact  that  the  Pythagoreans 
believed  in  the  transmigration  of  the  soul,  and,  for  that  reason, 
opposed  the  shedding  of  blood.  In  the  traditions  of  the  late 
Neo-Pythagoreans  the  objection  is  removed  by  replacing  the 
bloody  sacrifice  by  that  of  "  an  ox  made  of  flour  "  !  The  dem- 
onstration of  the  law  of  three  squares,  given  in  Euclid,  L,  47, 
is  due  to  Euclid  himself.  The  proof  given  by  Pythagoras 
has  not  been  handed  down  to  us.  Much  ingenuity  has  been 
expended  in  conjectures  as  to  its  nature.  Bretschneider's  sur- 
mise, that  the  Pythagorean  proof  was  substantially  the  same 
as  the  one  of  Bhaskara  (given  elsewhere),  has  been  well 
received  by  Hankel,  Allman,  Gow,  Loria.1  Cantor  thinks  it 
not  improbable  that  the  early  proof  involved  the  consideration 
of  special  cases,  of  which  the  isosceles 
right  triangle,  perhaps  the  first,  may 
have  been  proved  in  the  manner  indi- 
cated by  the  adjoining  figure.2  The 
four  lower  triangles  together  equal  the 
four  upper.  Divisions  of  squares  by  their 
diagonals  in  this  fashion  occur  in  Plato's 
Meno.  Since  the  time  of  the  Greeks  the  famous  Pythagorean 
Theorem  has  received  many  different  demonstrations.3 

A  characteristic  point  in  the  method  of  Pythagoras  and  his 
school  was  the  combination  of  geometry  and  arithmetic;    an 

1  See  BRETSCHNEIDER,  p.  82 ;  ALLMAN,  p.  36  ;  HANKEL,  p.  98 ;  Gow, 
p.  155 ;  LORIA,  I.,  p.  48. 

2  CANTOR,  L,  185;  see  also  Gow,  p.  155,  and  ALLMAN,  p.  29. 

3  See  JOH.  Jos.  JGN.  HOFFMANN,  Der  Pythagorische  Lehrsatz  mit  zwey 
und  dreysig  theils  bekannten,  theils  neuen  Beweisen.     Mainz,  1819.     See 
also  JURY  WIPPER,  Sechsundvierzig  Beweise  des  Pythagordischen  Lehr- 
satzes.     Aus  dem  Russischen  von  F.  GRAAP,  Leipzig,  1880.     The  largest 
collection  of  proofs  is  by  F.  Yanney  and  J.  A.  Calderhead  in  the  Am. 
Math.  Monthly,  Vols.  3  and  4,  1896  and  1897. 


GREECE  51 

arithmetical  fact  has  its  analogue  in  geometry,  and  vice  versa. 
Thus  in  connection  with  the  law  of  three  squares  Pythagoras 
devised  a  rule  for  finding  integral  numbers  representing  the 
lengths  of  the  sides  of  right  triangles :  Choose  2  n  +  1  as  one 
side,  then  £[(2w  +  I)2-  1]  =  2n2  +  2n=  the  other  side,  and 
2ft*  +  2n  +  l  =  the  hypotenuse.  If  n  =  5,  then  the  three 
sides  are  11,  60,  61.  This  rule  yields  only  triangles  whose 
hypotenuse  exceeds  one  of  the  sides  by  unity. 

Ascribed  to  the  Pythagoreans  is  one  of  the  greatest  mathe- 
matical discoveries  of  antiquity  —  that  of  Irrational  Quantities. 
The  discovery  is  usually  supposed  to  have  grown  out  of  the 
study  of  the  isosceles  right  triangle.1  If  each  of  the  equal 
legs  is  taken  as  unity,  then  the  hypotenuse,  being  equal 
to  V2,  cannot  be  exactly  represented  by  any  number  what- 
ever. We  may  imagine  that  other  numbers,  say  7  or  -J,  were 
taken  to  represent  the  legs ;  in  these  and  all  other  cases  experi- 
mented upon,  no  number  could  be  found  to  exactly  measure  the 
length  of  the  hypotenuse.  After  repeated  failures,  doubtless, 
"  some  rare  genius,  to  whom  it  is  granted,  during  some  happy 
moments,  to  soar  with  eagle's  flight  above  the  level  of  human 
thinking, — it  may  have  been  Pythagoras  himself, — grasped  the 
happy  thought  that  this  problem  cannot  be  solved."  2  As  there 
was  nothing  in  the  shape  of  any  geometrical  figure  which  could 
suggest  to  the  eye  the  existence  of  irrationals,  their  discovery 
must  have  resulted  from  unaided  abstract  thought.  The 
Pythagoreans  saw  in  irrationals  a  symbol  of  the  unspeakable. 
The  one  who  first  divulged  their  theory  is  said  to  have  suf- 
fered shipwreck  in  consequence,  "for  the  unspeakable  and 
invisible  should  always  be  kept  secret."3 

1  ALLMAN,  p.  42,  thinks  it  more  likely  that  the  discovery  was  owing  to 
the  problem  —  to  cut  a  line  in  extreme  and  mean  ratio. 

2  HANKEL,  p.  101. 

8  The  same  story  of  death  in  the  sea  is  told  of  the  Pythagorean  Hip- 
pasus  for  divulging  the  knowledge  of  the  dodecaedron. 


52  A    HISTORY    OF   MATHEMATICS 

The  theory  of  parallel  lines  enters  into  the  Pythagorean 
proof  of  the  angle-sum  in  triangles;  a  line  being  drawn  parallel 
to  the  base.  In  this  mode  of  proof  we  observe  progress  from 
the  special  to  the  general,  for  according  to  Geminus  the  early 
demonstration  (by  Thales?)  of  this  theorem  embraced  three 
different  cases:  that  of  equilateral,  that  of  isosceles,  and,  finally, 
that  of  scalene  triangles.1 

Eudemus  says  the  Pythagoreans  invented  the  problems  con- 
cerning the  application  of  areas,  including  the  cases  of  defect 
and  excess,  as  in  Euclid,  VI.,  28,  29.  They  could  also  construct 
a  polygon  equal  in  area  to  a  given  polygon  and  similar  to 
another.  In  a  general  way  it  may  be  said  that  the  Pythago- 
rean plane  geometry,  like  the  Egyptian,  was  much  concerned 
with  areas.  Conspicuous  is  the  absence  of  theorems  on  the 
circle. 

The  Pythagoreans  demonstrated  also  that  the  plane  about 
a  point  is  completely  filled  by  six  equilateral  triangles,  four 
squares,  or  three  regular  hexagons,  so  that  a  plane  can  be 
divided  into  figures  of  either  kind.  Related  to  the  study 
of  regular  polygons  is  that  of  the  regular  solids.  It  is  here 
that  the  Pythagoreans  contributed  to  solid  geometry.  From 
the  equilateral  triangle  and  the  square  arise  the  tetraedron, 
octaedron,  cube,  and  icosaedron  —  all  four  probably  known  to 
the  Egyptians,  certainly  the.  first  three.  In  Pythagorean  phi- 
losophy these  solids  represent,  respectively,  the  four  elements 
of  the  physical  world:  fire,  air,  earth,  and  water.  In  absence 
of  a  fifth  element,  the  subsequent  discovery  of  the  dodecaedron 
was  made  to  represent  the  universe  itself.  The  legend  goes 
that  Hippasus  perished  at  sea  because  he  divulged  "the  sphere 
with  the  twelve  pentagons." 

Pythagoras  used  to  say  that  the  most  beautiful  of  all  solids 
was  the  sphere ;  of  all  plane  figures,  the  circle. 

1  Consult  HANKEL,  pp.  95,  96. 


GREECE  53 

With  what  degree  of  rigour  the  Italian  school  demonstrated 
their  theorems  we  have  no  sure  way  of  determining.  We  are 
safe,  however,  in  assuming  that  the  progress  from  empirical  to 
reasoned  solutions  was  slow. 

Passing  to  the  later  Pythagoreans,  we  meet  the  name  of 
Philolaus,  who  wrote  a  book  on  Pythagorean  doctrines,  which 
he  made  known  to  the  world.  Lastly,  we  mention  the  brilliant 
Archytas  of  Tarentum  (428-347  B.C.),  who  was  the  only  great 
geometer  in  Greece  when  Plato  opened  his  school.  He  ad- 
vanced the  theory  of  proportion  and  wrote  on  the  duplication 
of  the  cube.1 

III.  The  Sophist  School.  —  The  periods  of  existence  of  the 
several  Greek  mathematical  "schools"  overlap  considerably. 
Thus,  Pythagorean  activity  continued  during  the  time  of  the 
sophists,  until  the  opening  of  the  Platonic  school. 

After  the  repulse  of  the  Persians  at  the  battle  of  Salamis 
in  480  B.C.  and  the  expulsion  of  the  Phoenicians  and  pirates 
from  the  ^Egean  Sea,  Greek  commerce  began  to  flourish.  Athens 
gained  great  ascendancy,  and  became  the  centre  toward  which 
scholars  gravitated.  Pythagoreans  flocked  thither;  Anaxag- 
oras  brought  to  Athens  Ionic  philosophy.  The  Pythagorean 
practice  of  secrecy  ceased  to  be  observed ;  the  spirit  of  Athe- 
nian life  demanded  publicity.2  All  menial  work  being  per- 
formed by  slaves,  the  Athenians  were  people  of  leisure.  That 
they  might  excel  in  public  discussions  on  philosophic  or  scien- 
tific questions,  they  must  be  educated.  There  arose  a  demand 
for  teachers,  who  were  called  sophists,  or  "  wise  men."  Unlike 
the  early  Pythagoreans,  the  sophists  accepted  pay  for  their 
teaching.  They  taught  principally  rhetoric,  but  also  philoso- 
phy, mathematics,  and  astronomy. 

The  geometry  of  the  circle,  neglected  by  the  Pythagoreans, 
was  now  taken  up.  The  researches  of  the  sophists  centre 

l  Consult  ALLMAN,  pp.  102-127.  2  ALLMAN,  p.  54. 


54  A   HISTORY  OF  MATHEMATICS 

around  the  three  following  famous  problems,  which  were  to  be 
constructed  with  aid  only  of  a  ruler  and  a  pair  of  compasses : 

(1)  The  trisection  of  any  angle  or  arc  ; 

(2)  The  duplication  of  the  cube ;   i.e.  to  construct  a  cube 
whose  volume  shall  be  double  that  of  a  given  cube; 

(3)  The  squaring  of  the  circle ;  i.e.  to  construct  a  square  or 
other  rectilinear  figure  whose  area  exactly  equals  the  area  of  a 
given  circle. 

Certainly  no  other  problems  in  mathematics  have  been  stud- 
ied so  assiduously  and  persistently  as  these.  The  best  Greek 
intellect  was  bent  upon  them  ;  Arabic  learning  was  applied  to 
them ;  some  of  the  best  mathematicians  of  the  Western  Renais- 
sance wrestled  with  them.  Trained  minds  and  untrained  minds, 
wise  men  and  cranks,  all  endeavoured  to  conquer  these  prob- 
lems which  the  best  brains  of  preceding  ages  had  tried  but  failed 
to  solve.  At  last  the  fact  dawned  upon  the  minds  of  men  that, 
so  long  as  they  limited  themselves  to  the  postulates  laid  down 
by  the  Greeks,  these  problems  did  not  admit  of  solution.  This 
divination  was  later  confirmed  by  rigorous  proof.  The  Greeks 
demanded  constructions  of  these  problems  by  ruler  and  com- 
passes, but  no  other  instruments.  In  other  words,  the  figure 
was  to  consist  only  of  straight  lines  and  of  circles.  A  con- 
struction was  not  geometrical  if  effected  by  drawing  ellipses, 
parabolas,  hyperbolas,  or  other  higher  curves.  With  aid  of 
such  curves  the  Greeks  themselves  resolved  all  three  problems; 
but  such  solutions  were  objected  to  as  mechanical,  whereby 
"  the  good  of  geometry  is  set  aside  and  destroyed,  for  we  again 
reduce  it  to  the  world  of  sense,  instead  of  elevating  and  imbu- 
ing it  with  the  eternal  and  incorporeal  images  of  thought,  even 
as  it  is  employed  by  God,  for  which  reason  He  always  is  God  " 
(Plato).  Why  should  the  Greeks  have  admitted  the  circle  into 
geometrical  constructions,  but  rejected  the  ellipse,  parabola, 
and  the  hyperbola  —  curves  of  the  same  order  as  the  circle  ? 


GREECE  55 

We  answer  in  the  words  of  Sir  Isaac  Newton  : l  "  It  is  not  the 
simplicity  of  the  equation,  but  the  easiness  of  the  description, 
which  is  to  determine  the  choice  of  our  lines  for  the  construc- 
tions of  problems.  For  the  equation  that  expresses  a  parabola 
is  more  simple  than  that  that  expresses  a  circle,  and  yet  the 
circle,  by  its  more  simple  construction,  is  admitted  before  it." 2 
The  bisection  of  an  angle  is  one  of  the  easiest  of  geometrical 
constructions.  Early  investigators,  as  also  beginners  in  our 
elementary  classes,  doubtless  expected  the  division  of  an  angle 
into  three  equal  parts  to  be  fully  as  easy.  In  the  special  case 
of  a  right  angle  the  construction  is  readily  found,  but  the 
general  case  offers  insuperable  difficulties.  One  Hippias,  very 
probably  Hippias  of  Elis  (born  about  460  B.C.),  was  among 
the  earliest  to  study  this  problem.  Failing  to  find  a  construc- 
tion involving  merely  circles  and  straight  lines,  he  discovered 
a  transcendental  curve  (i.e.  one  which  cannot  be  represented 
by  an  algebraic  equation)  by  which  an  angle  could  be  divided 
not  only  into  three,  but  into  any  number  of  equal  parts.  As 
this  same  curve  was  used  later  in  the  quadrature  of  the 
circle,  it  received  the  name  of  quadratrix.3 

1  ISAAC  NEWTON,  Universal  Arithmetick.     Translated  by  the  late  Mr. 
Ralphson ;  revised  by  Mr.  Cunn.     London,  1769,  p.  468. 

2  In  one  of  De  Morgan's  letters  to  Sir  W.  R.  Hamilton  occurs  the  fol- 
lowing: "But  what  distinguishes  the  straight  line  and  circle  more  than 
anything  else,  and  properly  separates  them  for  the  purpose  of  elementary 
geometry  ?    Their  self -similarity.    Every  inch  off  a  straight  line  coincides 
with  every  other  inch,  and  off  a  circle  with  every  other  off  the  same 
circle.    Where,  then,  did  Euclid  fail  ?    In  not  introducing  the  third  curve, 
which  has  the  same  property  —  the  screw.    The  right  line,  the  circle,  the 
screw  —  the  representatives  of  translation,  rotation,  and  the  two  combined 
—  ought  to  have  been  the  instruments  of  geometry.     With  a  screw  we 
should  never  have  heard  of  the  impossibility  of  trisecting  an  angle,  squar- 
ing the  circle,"  etc.  —  GRAVES,  Life  of  Sir  William  Eowan  Hamilton, 
1889,  Vol.  III.,  p.  343.    However,  if  Newton's  test  of  easiness  of  descrip- 
tion be  applied,  then  the  screw  must  be  excluded. 

8  For  a  description  of  the  quadratrix,  see  Gow,  p.  164.     That  the  in- 


66  A    HISTORY    OF    MATHEMATICS 

The  problem  "  to  double  the  cube  "  perhaps  suggested  itself 
to  geometers  as  an  extension  to  three  dimensions  of  the  prob- 
lem in  plane  geometry,  to  double  a  square.  If  upon  the  diag- 
onal of  a  square  a  new  square  is  constructed,  the  area  of  this 
new  square  is  exactly  twice  the  area  of  the  first  square.  This 
is  at  once  evident  from  the  Pythagorean  Theorem.  But  the 
construction  of  a  cube  double  in  volume  to  a  given  cube  brought 
to  light  unlooked-for  difficulties.  A  different  origin  is  assigned 
to  this  problem  by  Eratosthenes.  The  Delians  were  once  suf- 
fering from  a  pestilence,  and  were  ordered  by  the  oracle  to 
double  a  certain  cubical  altar.  Thoughtless  workmen  simply 
constructed  a  cube  with  edges  twice  as  long;  but  brainless 
work  like  that  did  not  pacify  the  gods.  The  error  being  dis- 
covered, Plato  was  consulted  on  this  "  Delian  problem."  Era- 
tosthenes tells  us  a  second  story :  King  Minos  is  represented 
by  an  old  tragic  poet  as  wishing  to  erect  a  tomb  for  his  son ; 
being  dissatisfied  with  the  dimensions  proposed  by  the  archi- 
tect, the  king  exclaimed :  "  Double  it,  but  fail  not  in  .  the 
cubical  form."  If  we  trust  these  stories,  then  the  problem 
originated  in  an  architectural  difficulty.1  Hippocrates  of  Chios 
(about  430  B.C.)  was  the  first  to  show  that  this  problem  can  be 
reduced  to  that  of  finding  between  a  given  line  and  another 
twice  as  long,  two  mean  proportionals,  i.e.  of  inserting  two 
lengths  between  the  lines,  so  that  the  four  shall  be  in  geomet- 
rical progression.  In  modern  notation,  if  a  and  2  a  are  the 
two  lines,  and  x  and  y  the  mean  proportionals,  we  have  the 

progression  a.  x,  y.  2  a,  which  gives  —  =  -  =  -^-;  whence  x2=  ay, 

x     y     2a 

y*=2ax.     Then  x4  =  a V  =  2  a?x,  xB  =  2as.     But  Hippocrates 

ventor  of  the  quadratrix  was  Hippias  of  Elis  is  denied  by  HANKEL,  p.  151, 
and  ALLMAN,  p.  94 ;  but  affirmed  by  CANTOR,  I.,  p.  181 ;  BRETSCHNEIDER, 
p.  94 ;  Gow,  p.  163  ;  LORIA,  I.,  p.  66 ;  TANNERY,  pp.  108,  131. 
1  Gow,  p.  162. 


GREECE  57 

naturally  failed  to  find  x,  the  side  of  the  double  cube,  by  geo- 
metrical construction.  However,  the  reduction  of  the  problem 
in  solid  geometry  to  one  of  plane  geometry  was  in  itself  no 
mean  achievement.  He  became  celebrated  also  for  his  success 
in  squaring  a  lune.  This  result  he  attempted  to  apply  to 
the  squaring  of  the  circle.1  In  his  study  of  the  Delian  and 
the  quadrature  problems  Hippocrates  contributed  much  to  the 
geometry  of  the  circle.  The  subject  of  similar  figures,  involv- 
ing the  theory  of  proportion,  also  engaged  his  attention.  He 
wrote  a  geometrical  text-book,  called  the  Elements  (now  lost), 
whereby,  no  doubt,  he  contributed  vastly  towards  the  progress 
of  geometry  by  making  it  more  easily  accessible  to  students. 

Hippocrates  is  said  to  have  once  lost  all  his  property. 
Some  accounts  say  he  fell  into  the  hands  of  pirates,  others 
attribute  the  loss  to  his  own  want  of  tact.  Says  Aristotle: 
"  It  is  well  known  that  persons  stupid  in  one  respect  are  by 
no  means  so  in  others.  There  is  nothing  strange  in  this :  so 
Hippocrates,  though  skilled  in  geometry,  appears  to  have  been 
in  other  respects  weak  and  stupid ;  and  he  lost,  as  they  say, 
through  his  simplicity,  a  large  sum  of  money  by  the  fraud  of 
the  collectors  of  customs  at  Byzantium."  2 

A  considerable  step  in  advance  was  the  introduction  of  the 
process  of  exhaustion  by  the  sophist  Antiphon,  a  contemporary 
of  Hippocrates.  By  inscribing  in  a  circle  a  square  and  on  its 
sides  erecting  isosceles  triangles  with  their  vertices  in  the  cir- 
cumference, and  on  the  sides  of  these  triangles  erecting  new 
isosceles  triangles,  etc.,  he  obtained  a  succession  of  regular  in- 
scribed polygons  of  8,  16,  32,  sides,  and  so  on,  each  polygon 
approaching  in  area  closer  to  that  of  the  circle  than  did  the 
preceding  polygon,  until  the  circle  was  finally  exhausted.  Anti- 

1  For  details  see  Gow,  pp.  165-168. 

2  Translated  by  ALLMAN,  p.  57,  from  Arist.  Eth.  ad  Eud.,VIl.,  c.  XIV.. 
p.  1247  a,  15,  ed.  Bekker. 


58  A   HISTORY    OF    MATHEMATICS 

phon  concluded  that  a  polygon  could  be  thus  inscribed,  the 
sides  of  which,  on  account  of  their  minuteness,  would  coincide 
with  the  circumference  of  the  circle.  Since  squares  can  be 
found  exactly  equal  in  area  to  any  given  polygon,  there  can 
be  constructed  a  square  exactly  equal  in  area  to  the  last  poly- 
gon inscribed,  and  therefore  equal  to  the  circle  itself.  Thus 
it  appears  that  he  claimed  to  have  established  the  possi- 
bility of  the  exact  quadrature  of  the  circle.  One  of  his  con- 
temporaries, Bryson  of  Heraclea,  modified  this  process  of 
exhaustion  by  not  only  inscribing,  but  also,  at  the  same 
time,  circumscribing  regular  polygons.  He  did  not  claim  to 
secure  coincidence  between  the  polygons  and  circle,  but  he 
committed  a  gross  error  by  assuming  the  area  of  the  circle  to 
be  the  exact  arithmetical  mean  of  the  two  polygonal  areas. 

Antiphon's  attempted  quadrature  involved  a  point  eagerly 
discussed  by  philosophers  of  that  time.  All  other  Greek 
geometers,  so  far  as  we  know,  denied  the  possibility  of  the 
coincidence  of  a  polygon  and  a  circle,  for  a  straight  line  can 
never  coincide  with  a  circumference  or  part  of  it.  If  a  poly- 
gon could  coincide  with  a  circle,  then,  says  Simplicius,  we 
would  have  to  put  aside  the  notion  that  magnitudes  are  divisi- 
ble ad  infinitum.  We  have  here  a  difficult  philosophical  ques- 
tion, the  discussion  of  which  at  Athens  appears  to  have 
greatly  influenced  and  modified  Greek  mathematical  thought 
in  respect  to  method.  The  Eleatic  philosophical  school,  with 
the  great  dialectician  Zeno  at  its  head,  argued  with  admirable 
ingenuity  against  the  infinite  divisibility  of  a  line,  or  other 
magnitude.  Zeno's  position  was  practically  taken  by  Antiphon 
in  his  assumption  that  straight  and  curved  lines  are  ultimately 
reducible  to  the  same  indivisible  elements.1  Zeno  reasoned  by 
reductio  ad  absurdum  against  the  theory  of  the  infinite  divisi- 

1  ALLMAN,  p.  56. 


GREECE  59 

bility  of  a  line.  He  argued  that  if  this  theory  is  assumed  to 
be  correct,  Achilles  could  not  catch  a  tortoise.  For  while  he 
ran  to  the  place  where  the  tortoise  had  been  when  he  started, 
the  tortoise  crept  some  distance  ahead,  and  while  Achilles 
hastened  to  that  second  spot,  it  again  moved  forward  a  little, 
and  so  on.  Being  thus  obliged  first  to  reach  every  one  of  the 
infinitely  many  places  which  the  tortoise  had  previously  occu- 
pied, Achilles  could  never  overtake  the  tortoise.  But,  as  a 
matter  of  fact,  Achilles  could  catch  a  tortoise,  therefore,  it  is 
wrong  to  assume  that  the  distance  can  be  divided  into  an 
indefinite  number  of  parts.  In  like  manner,  "  the  flying  arrow 
is  always  at  rest ;  for  it  is  at  each  moment  only  in  one  place." 
These  paradoxes  involving  the  infinite  divisibility,  and  there- 
fore the  infinite  multiplicity  of  parts  no  doubt  greatly  per- 
plexed the  mathematicians  of  the  time.  Desirous  of  construct- 
ing an  unassailable  geometric  structure,  they  banished  from 
their  science  the  ideas  of  the  infinitely  little  and  the  infinitely 
great.  Moreover,  to  meet  other  objections  of  dialecticians, 
theorems  evident  to  the  senses  (for  instance,  that  two  inter- 
secting circles  cannot  have  a  common  centre)  were  sub- 
jected to  rigorous  demonstration.  ,  Thus  the  influence  of 
dialecticians  like  Zeno,  themselves  not  mathematicians,  greatly 
modified  geometric  science  in  the  direction  of  increased  rigour.1 
The  process  of  exhaustion,  adopted  by  Antiphon  and  Bryson, 
was  developed  into  the  perfectly  rigorous  method  of  exhaustion. 
In  finding,  for  example,  the  ratio  between  the  areas  of  two 
circles,  similar  polygons  were  inscribed  and  by  increasing  the 
number  of  sides,  the  spaces  between  the  polygons  and  circles 
nearly  exhausted.  Since  the  polygonal  areas  were  to  each 
other  as  the  squares  of  the  diameters,  geometers  doubtless 
divined  the  theorem  attributed  to  Hippocrates  of  Chios,  that 

1  Consult  further,  HANKEL,  p.  118  ;  CANTOR,  I.,  p.  198  ;  ALLMAN,  p.  55 ; 
LORI  A,  I.,  p.  53. 


60  A   HISTORY   OF   MATHEMATICS 

the  circles  themselves  are  to  each  other  as  the  square  of  their 
diameters.  But  in  order  to  exclude  all  vagueness  or  possibility 
of  doubt,  later  Greek  geometers  applied  reasoning  like  that  in 
Euclid,  XII.,  2,  which  we  give  in  condensed  form  as  follows : 
Let  (7,  c  be  two  circular  areas;  Z>,  d,  the  diameters.  Then, 
if  the  proportion  D2 :  d?  =  C :  c  is  not  true,  suppose  that 
Z>2 :  d2  =  C :  c'.  If  c'  <  c,  then  a  polygon  p  can  be  inscribed  in 
the  circle  c  which  comes  nearer  to  c  in  area  than  does  c'.  If 
P  be  the  corresponding  polygon  in  (7,  then  P:p=D2:d?=C:c', 
and  P:C=p:c'.  Since  p>c',  we  have  P>  C,  which  is 
absurd.  Similarly,  c'  cannot  exceed  c.  As  c'  cannot  be  larger, 
nor  smaller  than  c,  it  must  be  equal  to  c.  Q.E.D.  Here  we  have 
exemplified  the  method  of  exhaustion,  involving  the  process  of 
reasoning,  designated  the  reductio  ad  absurdum.  Hankel  refers 
this  method  of  exhaustion  back  to  Hippocrates  of  Chios,  but 
the  reasons  for  assigning  it  to  this  early  writer,  rather  than 
to  Eudoxus,  seem  insufficient.1 

IV.  Tfie  Platonic  School.  —  After  the  Peloponnesian  War 
(431-404  B.C.),  the  political  power  of  Athens  declined,  but  her 
leadership  in  philosophy,  literature,  and  science  became  all  the 
stronger.  She  brought  forth  such  men  as  Plato  (429  ?-347  B.C.), 
the  strength  of  whose  mind  has  influenced  philosophical 
thought  of  all  ages.  Socrates,  his  early  teacher,  despised 
mathematics.  But  after  the  death  of  Socrates,  Plato  travelled 
extensively  and  came  in  contact  with  several  prominent  math- 
ematicians. At  Cyrene  he  studied  geometry  with  Theodorus ; 
in  Italy  he  met  the  Pythagoreans.  Archytas  of  Tarentum  and 
Timseus  of  Locri  became  his  intimate  friends.  About  389  B.C. 
Plato  returned  to  Athens,  founded  a  school  in  the  groves  of 
the  Academia,  and  devoted  the  remainder  of  his  life  to  teach- 
ing and  writing.  Unlike  his  master,  Socrates,  Plato  placed 

1  Consult  HANKEL,  p.  122  ;  Gow,  p.  173  j  CANTOR,  I.,  pp.  242,  247. 


GREECE  61 

great  value  upon  the  mind-developing  power  of  mathematics. 
"  Let  no  one  who  is  unacquainted  with  geometry  enter  here," 
was  inscribed  over  the  entrance  to  his  school.  Likewise  Xeno- 
crates,  a  successor  of  Plato,  as  teacher  in  the  Academy,  declined 
to  admit  a  pupil  without  mathematical  training,  "Depart,  for 
thou  hast  not  the  grip  of  philosophy."  The  Eudemian  Sum- 
mary says  of  Plato  that  "he  filled  his  writings  with  mathe- 
matical terms  and  illustrations,  and  exhibited  on  every  occa- 
sion the  remarkable  connection  between  mathematics  and 
philosophy." 

Plato  was  not  a  professed  mathematician.  He  did  little  or 
no  original  work,  but  he  encouraged  mathematical  study  and 
suggested  improvements  in  the  logic  and  methods  employed 
in  geometry.  He  turned  the  instinctive  logic  of  the  earlier 
geometers  into  a  method  to  be  used  consciously  and  without 
misgiving.1  With  him  begin  careful  definitions  and  the 
consideration  of  postulates  and  axioms.  The  Pythagorean 
definition,  "  a  point  is  unity  in  position,"  embodying  a  philo- 
sophical theory,  was  rejected  by  the  Plato nists;  a  point  was 
defined  as  "the  beginning  of  a  straight  line"  or  "an  indivisible 
line."  According  to  Aristotle  the  following  definitions  were 
also  current :  The  point,  the  line,  the  surface,  are  respectively 
the  boundaries  of  the  line,  the  surface,  and  the  solid ;  a  solid 
is  that  which  has  three  dimensions.  Aristotle  quotes  from 
the  Platonists  the  axiom :  If  equals  be  taken  from  equals,  the 
remainders  are  equal.  How  many  of  the  definitions  and  axi- 
oms were  due  to  Plato  himself  we  cannot  tell.  Proclus  and 
Diogenes  Laertius  name  Plato  as  the  inventor  of  the  method 
of  proof  called  analysis.  To  be  sure,  this  method  had  been 
used  unconsciously  by  Hippocrates  and  others,  but  it  is  gen- 
erally believed  that  Plato  was  the  one  who  turned  the  un- 

1  Gow,  pp.  175,  176.     See  also  HANKEL,  pp.  127-150. 


62  A   HISTORY   OF   MATHEMATICS 

conscious  logic  into  a  conscious,  legitimate  method.  The 
development  and  perfection  of  this  method  was  certainly  a 
great  achievement,  but  Allman  (p.  125)  is  more  inclined  to 
ascribe  it  to  Archytas  than  to  Plato. 

The  terms  synthesis  and  analysis  in  Greek  mathematics  had 
a  different  meaning  from  what  they  have  in  modern  mathe- 
matics or  in  logic.1  The  oldest  definition  of  analysis  as  op- 
posed to  synthesis  is  given  in  Euclid,  XIII.,  5,  which  was  most 
likely  framed  by  Eudoxus:2  "Analysis  is  the  obtaining  of 
the  thing  sought  by  assuming  it  and  so  reasoning  up  to  an 
admitted  truth ;  synthesis  is  the  obtaining  of  the  thing  sought 
by  reasoning  up  to  the  inference  and  proof  of  it."  3 

Plato  gave  a  powerful  stimulus  to  the  study  of  solid  geom- 
etry. The  sphere  and  the  regular  solids  had  been  studied 
somewhat  by  the  Pythagoreans  and  Egyptians.  The  latter, 

1  HANKEL,  pp.  137-150 ;   also  HANKEL,   Mathematik  in  den   letzten 
Jahrhunderten,  Tubingen,  1884,  p.  12. 

2  BRETSCHNEIDER,  p.  168. 

8  Greek  mathematics  exhibits  different  types  of  analysis.  One  is  the 
reductio  ad  absurdum,  in  the  method  of  exhaustion.  Suppose  we  wish 
to  prove  that  "A  is  B."  We  assume  that  A  is  not  B  ;  then  we  form  a 
synthetic  series  of  conclusions  :  not  B  is  (7,  C  is  D,  D  is  E ;  if  now  A  is 
not  E,  then  it  is  impossible  that  A  is  not  B  ;  i.e.,  A  is  B.  Q.E.D.  Verify 
this  process  by  taking  Euclid,  XII. ,  2,  given  above.  Allied  to  this  is  the 
theoretic  analysis :  To  prove  that  A  is  B  assume  that  A  is  B,  then  B  is  C, 
C  is  D,  D  is  E,  E  is  F;  hence  A  is  F.  If  this  last  is  known  to  be  false, 
then  A  is  not  B ;  if  it  is  known  to  be  true,  then  the  reasoning  thus  far 
is  not  conclusive.  To  remove  doubt  we  must  follow  the  reverse  process, 
A  is  F,  F  is  JE,  E  is  Z>,  D  is  O,  C  is  B ;  therefore  A  is  B.  This  second 
case  involves  two  processes,  the  analytic  followed  by  the  synthetic.  The 
only  aim  of  the  analytic  is  to  aid  in  the  discovery  of  the  synthetic.  Of 
greater  importance  to  the  Greeks  was  the  problematic  analysis,  applied 
in  constructions  intended  to  satisfy  given  conditions.  The  construction 
is  assumed  as  accomplished ;  then  the  geometric  relations  are  studied 
with  the  view  of  discovering  a  synthetic  solution  of  the  problem.  For 
examples  of  proofs  by  analysis,  consult  HANKEL,  p.  143 ;  Gow,  p.  178 ; 
ALLMAN,  pp.  160-163  ;  TODHUNTEB'S  Euclid,  1869,  Appendix,  pp.  320-328. 


GREECE  63 

of  course,  were  more  or  less  familiar  with  the  geometry  of 
the  pyramid.  In  the  Platonic  school,  the  prism,  pyramid, 
cylinder,  and  cone  were  investigated.  The  study  of  the  cone 
led  Mensechmus  to  the  discovery  of  the  conic  sections.  Per- 
haps the  most  brilliant  mathematician  of  this  period  was 
Eudoxus.  He  was  born  at  Cnidus  about  408  B.C.,  studied 
under  Archytas  and,  for  two  months,  under  Plato.  Later  he 
taught  at  Cyzicus.  At  one  time  he  visited,  with  his  pupils, 
the  Platonic  school.  He  died  at  Cyzicus  in  355  B.C.  Among 
the  pupils  of  Eudoxus  at  Cyzicus  who  afterwards  entered  the 
academy  of  Plato  were  Mensechmus,  Dinostratus,  Athenaeus, 
and  Helicon.  The  fame  of  the  Academy  is  largely  due  to 
these.  The  Eudemian  Summary  says  that  Eudoxus  "first 
increased  the  number  of  general  theorems,  added  to  the  three 
proportions  three  more,  and  raised  to  a  considerable  quantity 
the  learning,  begun  by  Plato,  on  the  subject  of  the  section,  to 
which  he  applied  the  analytic  method."  By  this  "section" 
is  meant,  no  doubt,  the  "golden  section,"  which  cuts  a  line 
in  extreme  and  mean  ratio.  He  proved,  says  Archimedes, 
that  a  pyramid  is  exactly  one-third  of  a  prism,  and  a  cone 
one-third  of  a  cylinder,  having  equal  base  and  altitude.  That 
spheres  are  to  each  other  as  the  cubes  of  their  radii,  was 
probably  established  by  him.  The  method  of  exhaustion 
was  used  by  him  extensively  and  was  probably  his  own  in- 
vention.1 

1  The  Eudemian  Summary  mentions,  besides  the  geometers  already 
named,  Thesetetus  of  Athens,  to  whom  Euclid  is  supposed  to  be  indebted 
in  the  composition  of  the  10th  book,  treating  of  incommensurables  (ALL- 
MAN,  pp.  206-215);  Leodamas  of  Thasos  ;  Neocleides  and  his  pupil  Leon, 
who  wrote  a  geometry  ;  Theudius  of  Magnesia,  who  also  wrote  a  geometry 
or  Elements  ;  Hermotimus  of  Colophon,  who  discovered  many  propositions 
in  Euclid's  Elements;  Amyclas  of  Heraclea,  Cyzicenus  of  Athens,  and 
Philippus  of  Mende.  The  pre-Euclidean  text-books  just  mentioned  are 
not  extant. 


64  A   HISTORY   OF   MATHEMATICS 

V.  The  First  Alexandrian  School.  —  During  the  sixty-six 
years  following  the  Peloponnesian  War  —  a  period  of  political 
decline  —  Athens  brought  forth  some  of  the  greatest  and  most 
subtle  thinkers  of  Greek  antiquity.  In  338  B.C.  she  was  con- 
quered by  Philip  of  Macedon  and  her  power  broken  forever. 
Soon  after,  Alexandria  was  founded  by  Alexander  the  Great, 
and  it  was  in  this  city  that  literature,  philosophy,  science, 
and  art  found  a  new  home/ 

In  the  course  of  our  narrative,  we  have  seen  geometry  take 
feeble  root  in  Egypt;  we  have  seen  it  transplanted  to  the 
Ionian  Isles ;  thence  to  Lower  Italy  and  to  Athens ;  now,  at 
last,  grown  to  substantial  and  graceful  proportions,  we  see  it 
transferred  to  the  land  of  its  origin,  and  there,  newly  invigor- 
ated, expand  in  exuberant  growth. 

Perhaps  the  founder,  certainly  a  central  figure,  of  the  Alex- 
andrian mathematical  school  was  Euclid  (about  300  B.C.  ).  No 
ancient  writer  in  any  branch  of  knowledge  has  held  such  a  com- 
manding position  in  modern  education  as  has  Euclid  in  ele- 
mentary geometry.  "  The  sacred  writings  excepted,  no  Greek 
has  been  so  much  read  or  so  variously  translated  as  Euclid." * 

After  mentioning  Eudoxus,  Theaetetus,  and  other  members 
of  the  Platonic  school,  Proclus 2  adds  the  following  to  the  Eu- 
demian  Summary : 

"Not  much  later  than  these  is  Euclid,  who  wrote  the  Ele- 
ments, arranged  much  of  Eudoxus's  work,  completed  much 
of  Thesetetus's,  and  brought  to  irrefragable  proof  propositions 
which  had  been  less  strictly  proved  by  his  predecessors. 
Euclid  lived  during  the  reign  of  the  first  Ptolemy,  for  he  is 
quoted  by  Archimedes  in  his  first  book ;  and  it  is  said,  more- 
over, that  Ptolemy  once  asked  him,  whether  in  geometric  mat- 

1  DE  MORGAN,  "  Eucleides  "  in  Smith's  Dictionary  of  Greek  and  Roman 
Biography  and  Mythology.     We  commend  this  remarkable  article  to  all. 

2  PROCLUS  (Ed.  FRIEDLEIN),  p.  68. 


GREECE  65 

ters  there  was  not  a  shorter  path  than  through  his  Elements, 
to  which  he  replied  that  there  was  no  royal  road  to  geometry.1 
He  is,  therefore,  younger  than  the  pupils  of  Plato,  but  older 
than  Eratosthenes  and  Archimedes,  for  these  are  contempora- 
ries, as  Eratosthenes  informs  us.  He  belonged  to  the  Platonic 
sect  and  was  familiar  with  Platonic  philosophy,  so  much  so,  in 
fact,  that  he  set  forth  the  final  aim  of  his  work  on  the  Elements 
to  be  the  construction  of  the  so-called  Platonic  figures  (regular 
solids)." 2  Pleasing  are  the  remarks  of  Pappus,3  who  says  he 
was  gentle  and  amiable  to  all  those  who  could  in  the  least 
degree  advance  mathematical  science.  Stobseus4  tells  the  fol- 
lowing story  :  "  A  youth  who  had  begun  to  read  geometry  with 
Euclid,  when  he  had  learned  the  first  proposition,  inquired, 
'  What  do  I  get  by  learning  those  things  ?  '  So  Euclid  called 
his  slave  and  said,  '  Give  him  threepence,  since  he  must  gain 
out  of  what  he  learns.7 " 

Not  much  more  than  what  is  given  in  these  extracts  do  we 
know  concerning  the  life  of  Euclid.  All  other  statements 
about  him  are  trivial,  of  doubtful  authority,  or  clearly 
erroneous.5 


1  "This  piece  of  wit  has  had  many  imitators.    'Quel  diable,'  said  a 
French  nobleman  to  Rohault,  his  teacher  in  geometry,  '  pourrait  entendre 
cela  ? '  to  which  the  answer  was,  *  Ce  serait  un  diable  qui  aurait  de  la 
patience.'     A  story  similar  to  that  of  Euclid  is  related  by  Seneca  (Ep.  91, 
cited  by  August)  of  Alexander."     DE  MORGAN,  op.  cit. 

2  This  statement  of  Euclid's  aim  is  obviously  erroneous. 
8  PAPPUS  (Ed.  HULTSCH),  pp.  676-678. 

4  Quoted  by  Gow,  p.  195  from  Floril.  IV.,  p.  250. 

5  Syrian  and  Arabian  writers  claim  to  possess  more  information  about 
Euclid  ;  they  say  that  his  father  was  Naucrates,  that  Euclid  was  a  Greek 
born  in  Tyre,  that  he  lived  in  Damascus  and  edited  the  Elements  of  Apol- 
lonius.     For  extracts  from  Arabic  authors  and  for  a  list  of  books  on  the 
principal  editions  of  Euclid,  see  LORIA,  II.,  pp.  10,  11,  17,  18.      During 
the  middle  ages  the  geometer  Euclid  was  confused  with  Euclid  of  Megara, 
a  pupil  of  Socrates.     Of  interest  is  the  following  quotation  from  DE  MOR- 


66  A   HISTORY   OF    MATHEMATICS 

Though  the  author  of  several  works  on  mathematics  and 
physics,  the  fame  of  Euclid  has  at  all  times  rested  mainly  upon 
his  book  on  geometry,  called  the  Elements.  This  book  was  so 
far  superior  to  the  Elements  written  by  Hippocrates,  Leon,  and 
Theudius,  that  the  latter  works  soon  perished  in  the  struggle 
for  existence.  The  great  role  that  it  has  played  in  geometric 
teaching  during  all  subsequent  centuries,  as  also  its  strong  and 
weak  points,  viewed  in  the  light  of  pedagogical  science  and  of 
modern  geometrical  discoveries,  will  be  discussed  more  fully 
later.  At  present  we  confine  ourselves  to  a  brief  critical 
account  of  its  contents. 

Exactly  how  much  of  the  Elements  is  original  with  Euclid, 
we  have  no  means  of  ascertaining.  Positive  we  are  that  certain 
early  editors  of  the  Elements  were  wrong  in  their  view  that 
a  finished  and  unassailable  system  of  geometry  sprang  at  once 
from  the  brain  of  Euclid,  "  an  armed  Minerva  from  the  head 
of  Jupiter."  Historical  research  has  shown  that  Euclid  got 
the  larger  part  of  his  material  from  the  eminent  mathema- 
ticians who  preceded  him.  In  fact,  the  proof  of  the  "  Theorem 
of  Pythagoras"  is  the  only  one  directly  ascribed  to  him. 
Allman1  conjectures  that  the  substance  of  Books  I.,  II.,  IV. 
comes  from  the  Pythagoreans,  that  the  substance  of  Book  VI. 
is  due  to  the  Pythagoreans  and  Eudoxus,  the  latter  contribut- 
ing the  doctrine  of  proportion  as  applicable  to  incommensu- 
rables  and  also  the  Method  of  Exhaustions  (Book  XII.),  that 
Theaetetus  contributed  much  toward  Books  X.  and  XIII.,  that 
the  principal  part  of  the  original  work  of  Euclid  himself  is  to 

GAN,  op.  cit.:  "  In  the  frontispiece  to  Whiston's  translation  of  Tacquet's 
Euclid  there  is  a  bust,  which  is  said  to  be  taken  from  a  brass  coin  in  pos- 
session of  Christina  of  Sweden  ;  but  no  such  coin  appears  in  the  published 
collection  of  those  in  the  cabinet  of  the  queen  of  Sweden.  Sidonius  Apol- 
linaris  says  (Epist.  XI.,  9)  that  it  was  the  custom  to  paint  Euclid  with 
the  fingers  extended  (laxatis),  as  if  in  the  act  of  measurement." 
1  ALLMAN,  pp.  211,  212. 


GREECE  67 

be  found  in  Book  X.  The  greatest  achievement  of  Euclid,  no 
doubt,  was  the  co-ordinating  and  systematizing  of  the  material 
handed  down  to  him.  He  deserves  to  be  ranked  as  one  of  the 
greatest  systematizers  of  all  time. 

"The  contents  of  the  Elements  may  be  briefly  indicated  as 
follows :  Books  I.,  II.,  III.,  IV.,  VI.  treat  of  plane  geometry ; 
Book  V.,  of  the  theory  of  proportion  applicable  to  magnitudes 
in  general ;  Books  VII.,  VIII.,  IX.,  of  arithmetic ;  Book  X.,  of 
the  arithmetical  characteristics  of  divisions  of  straight  lines 
(i.e.  of  irrationals)  ;  Books  XI.,  XII.,  of  solid  geometry  j  Books 
XIII.,  XIV.,  XV.,  of  the  regular  solids.  The  last  two  books 
are  apocryphal,  and  are  supposed  to  have  been  written  by 
Hypsicles  and  Damascius,  respectively.1 

Difference  of  opinion  long  existed  regarding  the  merits  of  the 
Elements  as  a  scientific  treatise.  Some  regarded  it  as  a  work 
whose  logic  is  in  every  detail  perfect  and  unassailable,  while 
others  pronounced  it  to  be  "  riddled  with  fallacies." 2  In  our 
opinion,  neither  view  is  correct.  That  the  text  of  the  Elements 
is  not  free  from  faults  is  evident  to  any  one  who  reads  the  com- 
mentators on  Euclid.  Perhaps  no  one  ever  surpassed  Robert 
Simson  in  admiration  for  the  great  Alexandrian.  Yet  Sim- 
son's  "notes"  to  the  text  disclose  numerous  defects.  While 
Simson  is  certainly  wrong  in  attributing  to  blundering  editors 
all  the  defects  which  he  noticed  in  the  Elements,  early  editors 
are  doubtless  responsible  for  many  of  them.  Most  of  the 
emendations  pertain  to  points  of  minor  importance.  The 
work  as  a  whole  possesses  a  high  standard  of  accuracy.  A 
minute  examination  of  the  text  has  disclosed  to  commentators 

1  See  Gow,  p.  272 ;  CANTOR,  I.,  pp.  358,  501.  Book  XV.,  in  the  opinion 
of  Heiberg  and  others,  consists  of  three  parts,  of  which  the  third  is  perhaps 
due  to  Damascius.    See  LORIA,  II.,  pp.  88-92. 

2  See  C.  S.  PEIRCE  in  Nation,  Vol.  54,  1892,  pp.  116,  366,  and  in  the 
Monist,  July,  1892,  p.  539  j  G.  B.  HALSTED,  Educational  Review,  8,  1894, 
pp.  91-93. 


68  A   HISTORY   OF   MATHEMATICS 

an  occasional  lack  of  extreme  precision  in  the  statement  of 
what  is  assumed  without  proof,  for  truths  are  treated  as 
self-evident  which  are  not  found  in  the  list  of  postulates.1 
Again,  Euclid  sometimes  assumes  what  might  be  proved;  as 
when  in  the  very  definitions  he  asserts  that  the  diameter  of  a 
circle  bisects  the  figure,2  which  might  be  readily  proved  from 
the  axioms.  He  defines  a  plane  angle  as  the  "  inclination  of 
two  straight  lines  to  one  another,  which  meet  together,  but  are 
not  in  the  same  straight  line,"  but  leaves  the  idea  of  angle 
magnitude  somewhat  indefinite  by  his  failure  to  give  a  test  for 
equality  of  two  angles  or  to  state  what  constitutes  the  sum  or 
difference  of  two  angles.3  Sometimes  Euclid  fails  to  consider 
or  give  all  the  special  cases  necessary  for  the  full  and  complete 
proof  of  a  theorem.4  Such  instances  of  defects  which  friendly 
critics  have  found  in  the  Elements  show  that  Euclid  is  not 
infallible.5  But  in  noticing  these  faults,  we  must  not  lose 

1  For  instance,  the  intersection  of  the  circles  in  I.,  1,  and  in  I.,  22. 
See  also  H.  M.  TAYLOR'S  Euclid,  1893,  p.  vii. 

2  DE  MORGAN,  article  "  Euclid  of  Alexandria,"  in  the  English  Cyclo- 
pcedia. 

8  See  SIMON  NEWCOMB,  Elements  of  Geometry,  1884,  Preface  ;  H.  M. 
TAYLOR'S  Euclid,  p.  8 ;  DE  MORGAN,  The  Connexion  of  Number  and 
Magnitude,  London,  1836,  p.  85. 

4  Consult  TODHUNTER'S  Euclid,  notes  on  I.,  35;  III.,  21;  XL,  21. 
SIMSON'S  Euclid,  notes  on  L,  7  ;  III.,  35. 

6  A  proof  which  has  been  repeatedly  attacked  is  that  of  L,  16,  which 
"  uses  no  premises  not  as  true  in  the  case  of  spherical  as  in  that  of  plane 
triangles  ;  and  yet  the  conclusion  drawn  from  these  premises  is  known  to 
be  false  of  spherical  triangles."  See  Nation,  Vol.  54,  pp.  116,  366; 
E.  T.  DIXON,  in  Association  for  the  Improvement  of  Geometrical  Teaching 
(A.  I.  G.  T.),  17th  General  Report,  1891,  p.  29;  ENGEL  und  STACKEL, 
Die  Theorie  der  Parallellinien  von  Euklid  bis  auf  Gauss,  Leipzig,  1895, 
p.  11,  note.  (Hereafter  we  shall  cite  this  book  as  ENGEL  and  STACKEL.) 
In  the  Monist,  July,  1894,  p.  485,  G.  B.  Halsted  defends  the  proof  of  L,  16, 
arguing  that  spherics  are  ruled  out  by  the  postulate,  "Two  straight  lines 
cannot  enclose  a  space."  If  we  were  sure  that  Euclid  used  this  postulate, 
then  the  proof  of  L,  16  would  be  unobjectionable,  but  it  is  probable  that 


GREECE  69 

sight  of  the  general  excellence  of  the  work  as  a  scientific 
treatise  —  an  excellence  which  in  1877  was  fittingly  recog- 
nized by  a  committee  of  the  British  Association  for  the 
Advancement  of  Science  (comprising  some  of  England's  ablest 
mathematicians)  in  their  report  that  "  no  text-book  that  "has 
yet  been  produced  is  fit  to  succeed  Euclid  in  the  position  of 
authority." l  As  already  remarked,  some  editors  of  the  Elements, 
particularly  Eobert  Simson,  wrote  on  the  supposition  that  the 
original  Euclid  was  perfect,  and  any  defects  in  the  text  as 
known  to  them,  they  attributed  to  corruptions.  For  example, 
Simson  thinks  there  should  be  a  definition  of  compound  ratio 
at  the  beginning  of  the  fifth  book ;  so  he  inserts  one  and  assures 
us  that  it  is  the  very  definition  given  by  Euclid.  Not  a  single 
manuscript,  however,  supports  him.  The  text  of  the  Elements 
now  commonly  used  is  Theon's.  Simson  was  inclined  to  make 
him  the  scapegoat  for  all  defects  which  he  thought  he  discov- 
ered in  Euclid.  But  a  copy  of  the  Elements  sent  with  other 
manuscripts  from  the  Vatican  to  Paris  by  Napoleon  I.  is 
believed  to  be  anterior  to  Theon's  recension;  this  differs  but 
slightly  from  Theon's  version,  showing  that  the  faults  are 
probably  Euclid's  own. 

At  the  beginning  of  our  modern  translations  of  the  Elements 
(Eobert  Simson's  or  Todhunter's,  for  example),  under  the  head 
of  definitions  are  given  the  assumptions  of  such  notions  as  the 
point,  line,  etc.,  and  some  verbal  explanations.  Then  follow 
three  postulates  or  demands,  (1)  that  a  line  may  be  drawn  from 
any  point  to  any  other,  (2)  that  a  line  may  be  indefinitely  pro- 
duced, (3)  that  a  circle  may  be  drawn  with  any  radius  and  any 
point  as  centre.  After  these  come  twelve  axioms.2  The  term 

the  postulate  was  omitted  by  Euclid  and  supplied  by  some  commentator. 
See  HEIBERG,  Euclidis  Elementa. 

1  A.  I.  G.  T.,  6th  General  Report,  1878,  p.  14. 

a  Of  these  twelve  "  axioms,"  five  are  supposed  not  to  have  been  given  by 


70  A   HISTORY   OF   MATHEMATICS 

axiom  was  used  by  Proclus,  but  not  by  Euclid.  He  speaks 
instead  of  "  common  notions  "  —  common  either  to  all  men  or  to 
all  sciences.  The  first  nine  "  axioms  "  relate  to  all  kinds  of  mag- 
nitudes (things  equal  to  the  same  thing  are  equal  to  each  other, 
etc.,  the  whole  is  greater  than  its  part),1  while  the  last  three 
(two  straight  lines  cannot  enclose  a  space  ;  all  right  angles  are 
equal  to  one  another ;  the  parallel-axiom)  relate  to  space  only. 
While  in  nearly  all  but  the  most  recent  of  modern  editions 
of  Euclid  the  geometric  "axioms"  are  placed  in  the  same 
category  with  the  other  nine,  it  is  certainly  true  that  Euclid 
sharply  distinguished  between  the  two  classes.  An  immense 
preponderance  of  manuscripts  places  the  "  axioms  "  relating  to 
space  among  the  postulates.2  This  is  their  proper  place,  for 
modern  research  has  shown  that  they  are  assumptions  and 
not  common  notions  or  axioms.  It  is  not  known  who  first 
made  the  unfortunate  change.  In  this  respect  there  should 

Euclid,  viz.  the  four  on  inequalities  and  the  one,  "  two  straight  lines  can- 
not enclose  a  space."  Thus  HEIBERG  and  MENGE,  in  their  Latin  and 
Greek  edition  of  1883,  omit  all  five.  See  also  ENGEL  and  STACKEL, 
p.  8,  note. 

1  "That the  whole  is  greater  than  its  part  is  not  an  axiom,  as  that  emi- 
nently bad  reasoner,  Euclid,  made  it  to  be Of  finite  collections  it  is  true, 

of  infinite  collections  false."  —  C.  S.  PEIRCE,  Monist,  July,  1892,  p.  539. 
Peirce  gives  illustrations  in  which,  for  infinite  collections,  the  "  axiom  "  is 
untrue.     Nevertheless,  we  are  unwilling  to  admit  that  the  assuming  of 
this  axiom  proves  Euclid  a  bad  reasoner.     Euclid  had  nothing  whatever 
to  do  with  infinite  collections.     As  for  finite  collections,  it  would  seem 
that  nothing  can  be  more  axiomatic.     To  infinite  collections  the  terms 
great  and  small  are  inapplicable.     On  this  point  see  GEORG   CANTOR, 
"Ueber  eine  Eigenschaft  des  Inbegriffes  reeller  algebraischer  Zahlen," 
Crelle  Journal,  77,  1873  ;  or  for  a  more  elementary  discussion,  see  FELIX 
KLEIN,  Ausgewahlte  Fragen  der  Elementargeometrie,  ausgearbeitet  von 
F.  TAGERT,  Leipzig,  1895,  p.  39.     [This  work  will  be  cited  hereafter  as 
KLEIN.]     That  "the  whole  is  greater  than  its  part"  does  not  apply  in 
the  comparison  of  infinites  was  recognized  by  Bolyai.     See  HALSTED'S 
BOLYAI'S  Science  Absolute  of  Space,  4th  Ed.,  1896,  §  24,  p.  20. 

2  HANKEL,  Die  Complexen  Zahlen,  Leipzig,  1867,  p.  52. 


GREECE  71 

be  a  speedy  return  to  Euclid's  practice.  The  parallel-postu- 
late plays  a  very  important  role  in  the  history  of  geometry.1 
Most  modern  authorities  hold  that  Euclid  missed  one  of  the 
postulates,  that  of  rigidity  (or  else  that  of  equal  variation), 
which  demands  that  figures  may  be  moved  about  in  space 
without  any  alteration  in  form  or  magnitude  (or  that  all  mov- 
ing figures  change  equally  and  each  fills  the  same  space  when 
brought  back  to  its  original  position).2  The  rigidity  postulate 
is  given  in  recent  geometries,  but  a  contradictory  of  it,  stated 
above  (the  postulate  of  equal  variation)  would  likewise  admit 
figures  to  be  compared  by  the  method  of  superposition,  and 
"everything  would  go  on  quite  as  well"  (Clifford).  Of  the 
two,  the  former  is  the  simpler  postulate  and  is,  moreover,  in 
accordance  with  what  we  conceive  to  be  our  every-day  experi- 
ence. G.  B.  Halsted  contends  that  Euclid  did  not  miss  the 
rigidity  assumption  and  is  justified  in  not  making  it,  since  he 
covers  it  by  his  assumption  8 :  "  Magnitudes  which  can  be 
made  to  coincide  with  one  another  are  equal  to  one  another." 

In  the  first  book,  Euclid  only  once  imagines  figures  to  be 
moved  relatively  to  each  other,  namely,  in  proving  proposi- 

1  The  parallel-postulate  is  :  "  If  a  straight  line  meet  two  straight  lines, 
so  as  to  make  the  two  interior  angles  on  the  same  side  of  it  taken  to- 
gether less  than  two  right  angles,  these  straight  lines,  being  continually 
produced,  shall  at  length  meet  on  that  side  on  which  are  the  angles  which 
are  less  than  two  right  angles."    In  the  various  editions  of  Euclid,  different 
numbers  are  assigned  to  the  "axioms."     Thus,  the  parallel-postulate  is 
in  old  manuscripts  the  5th  postulate.     This  place  is  also  assigned  to  it  by 
F.  PEYRARD  (who  was  the  first  to  critically  compare  the  various  MSS.) 
in  his  edition  of  Euclid  in  French  and  Latin,  1814,  and  by  HEIBERG  and 
MENGE  in  their  excellent  annotated  edition  of  Euclid's  works,  in  Greek 
and  Latin,  Leipzig,  1883.    Clavius  calls  it  the  13th  axiom ;  Robert  Sim- 
son,  the  12th  axiom;  others  (Bolyai  for  instance),  the  llth  axiom. 

2  Consult  W.  K.  CLIFFORD,  The  Common  Sense  of  the  Exact  Sciences, 
1885,  p.  54.     "  (1)  Different  things  changed  equally,  and  (2)  anything 
which  was  carried  about  and  brought  back  to  its  original  position  filled 
the  same  space." 


72  A   HISTORY   OF   MATHEMATICS 

tion  4 :  two  triangles  are  equal  if  two  sides  and  the  included 
angle  are  equal  respectively.  To  bring  the  triangles  into  coin- 
cidence, one  triangle  may  have  to  be  turned  over,  but  Euclid 
is  silent  on  this  point.  "  Can  it  have  escaped  his  notice  that 
in  plane  geometry  there  is  an  essential  difference  between 
motion  of  translation  and  reversion  ?  " l 

Book  V.,  on  proportions  of  magnitudes,  has  been  greatly 
admired  because  of  its  rigour  of  treatment.2  Beginners  find 
the  book  difficult.  It  has  been  the  chief  battle-ground  of  dis- 
cussion regarding  the  fitness  of  the  Elements  as  a  text-book 
for  beginners. 

Book  X.  (as  also  Books  VII.,  VIII.,  IX.,  XIII.,  XIV.,  XV.) 
is  omitted  from  modern  school  editions.  But  it  is  the  most 
wonderful  book  of  all.  Euclid  investigates  every  possible 

variety  of  lines  which  can  be  represented  by  \  V«  ±  V&, 
a  and  b  representing  two  commensurable  lines,  and  obtains  25 
species.  Every  individual  of  every  species  is  incommensu- 
rable with  all  the  individuals  of  every  other  species.  De 
Morgan  was  enthusiastic  in  his  admiration  of  this  book.3 

1  ENGEL  and  STACKEL,  p.  8,  note. 

2  For  interesting  comments  on  Books  V.  and  VI.,  see  HANKEL,  pp. 
389-404. 

8  See  his  articles  "Eucleides"  in  SMITH'S  Die.  of  Greek  and  Eoman 
Biog.  and  Myth,  and  "Irrational  Quantity"  in  the  Penny  Cyclopaedia 
or  in  the  English  Cyclopaedia.  See  also  NESSELMANN,  pp.  165-183.  In 
connection  with  this  subject  of  irrationals  a  remark  by  Dedekind  is  of 
interest.  See  RICHARD  DEDEKIND,  Was  sind  und  was  sollen  die  Zahlen, 
Braunschweig,  1888,  pp.  xii  and  xiii.  He  points  out  that  all  Euclid's 
constructions  of  figures  could  be  made,  even  if  the  plane  were  not  con- 
tinuous ;  that  is,  even  if  certain  points  in  the  plane  were  imagined  to  be 
punched  out,  so  as  to  give  it  the  appearance  of  a  sieve.  All  the  points 
in  Euclid's  constructions  would  lie  between  the  holes ;  no  point  of  the 
constructions  would  fall  into  a  hole.  The  explanation  of  all  this  is  to  be 
sought  in  the  fact  that  Euclid  deals  with  certain  algebraic  irrationals,  to 
the  exclusion  of  the  transcendental. 


GREECE  73 

The  main  differences  in  subject-matter  between  the  Elements 
and  our  modern  school  geometries  consist  in  this :  the  modern 
works  pay  less  attention  to  the  "Platonic  figures,"  but  add 
theorems  on  triangles  and  quadrilaterals  inscribed  in  or  cir- 
cumscribed about  a  circle,  on  the  centre  of  gravity  of  the  tri- 
angle, on  spherical  triangles  (in  general,  the  geometry  of  a 
spherical  surface),  and,  perhaps,  on  some  of  the  more  recent 
discoveries  pertaining  to  the  geometry  of  the  plane  triangle 
and  circle. 

The  main  difference  as  to  method,  between  Euclid  and  his 
modern  rivals,  lies  in  the  treatment  of  proportion  and  the 
development  of  size-relations.  His  geometric  theory  of  pro- 
portion enables  him  to  study  these  relations  without  reference 
to  mensuration.  The  Elements  and  all  Greek  geometry  before 
Archimedes  eschew  mensuration.  The  theorem  that  the  area 
of  a  triangle  equals  half  the  product  of  its  base  and  its  altitude, 
or  that  the  area  of  a  circle  equals  IT  times  the  square  of  the 
radius,  is  foreign  to  Euclid.  In  fact  he  nowhere  finds  an 
approximation  to  the  ratio  between  the  circumference  and 
diameter.  Another  difference  is  that  Euclid,  unlike  the  great 
majority  of  modern  writers,  never  draws  a  line  or  constructs  a 
figure  until  he  has  actually  shown  the  possibility  of  such  con- 
struction with  aid  only  of  the  first  three  of  his  postulates 
or  of  some  previous  construction.  The  first  three  propositions 
of  Book  I.  are  not  theorems,  but  problems,  (1)  to  describe  an 
equilateral  triangle,  (2)  from  a  given  point  to  draw  a  straight 
line  equal  to  a  given  straight  line,  (3)  from  the  greater  of  two 
straight  lines  to  cut  off  a  part  equal  to  the  less.  It  is  only  by 
the  use  of  hypothetical  figures  that  modern  books  can  relegate 
all  constructions  to  the  end  of  chapters.  For  instance,  an 
angle  is  imagined  to  be  bisected,  before  the  possibility  and 
method  of  bisecting  it  has  been  shown.  One  of  the  most 
startling  examples  of  hypothetical  constructions  is  the  division 


74  A   HISTORY   OF   MATHEMATICS 

of  a  circumference  into  any  desired  number  of  equal  parts, 
given  in  a  modern  text-book.  This  appears  all  the  more  star- 
tling, when  we  remember  that  one  of  the  discoveries  which 
make  the  name  of  Gauss  immortal  is  the  theorem  that,  besides 
regular  polygons  of  2n,  3,  5  sides  (and  combinations  therefrom), 
only  polygons  whose  number  of  sides  is  a  prime  number  larger 
than  five,  and  of  the  form  p  =  2*n  -f  1,  can  be  inscribed  in  a 
circle  with  aid  of  Euclid's  postulates,  i.e.  with  aid  of  ruler  and 
compasses  only.1  Is  the  absence  of  hypothetical  constructions 
commendable  ?  If  the  aim  is  rigour,  we  answer  emphatically, 
Yes.  If  the  aim  is  adherence  to  recognized  pedagogical  prin- 
ciples, then  we  answer,  in  a  general  way,  that,  in  the  transition 
from  the  concrete  to  the  more  abstract  geometry,  it  often  seems 
desirable  to  let  facts  of  observation  take  the  place  of  abstruse 
processes  of  reasoning.  Even  Euclid  resorts  to  observation  for 
the  fact  that  his  two  circles  in  I.,  1  cut  each  other.2  Reasoning 
too  difficult  to  be  grasped  does  not  develop  the  mind.3  More- 
over, the  beginner,  like  the  ancient  Epicureans,  takes  no  interest 
in  trains  of  reasoning  which  prove  to  him  what  he  has  long 
known;  geometrical  reasoning  is  more  apt  to  interest  him 
when  it  discloses  new  facts.  Thus,  pedagogics  may  reasona- 
bly demand  some  concessions  from  demonstrative  rigour. 

Of  the  other  works  of  Euclid  we  mention  only  the  Data, 
probably  intended  for  students  who  had  completed  the  Ele- 
ments and  wanted  drill  in  solving  new  problems ;  a  lost  work 
on  Fallacies,  containing  exercises  in  detecting  fallacies;  a 
treatise  on  Porisms,  also  lost,  but  restored  by  Eobert  Simson 
and  Michel  Chasles. 

1  KLEIN,  p.  2. 

2  The  Epicureans,  says  Proclus,  blamed  Euclid  for  proving  some  things 
which  were  evident  without  proof.      Thus,  they  derided  I.,  20  (two  sides 
of  a  triangle  are  greater  than  the  third)  as  being  manifest  even  to  asses. 

8  For  discussions  of  the  subject  of  Hypothetical  Constructions,  see 
E.  L.  RICHARDS  in  Educat.  Review,  Vol.  III.,  1892,  p.  34  ;  G.  B.  HALSTED 
in  same  journal,  Vol.  IV.,  1893,  p.  152. 


GREECE  75 

The  period  in  which  Euclid  flourished  was  the  golden  era  in 
Greek  mathematical  history.  This  era  brought  forth  the  two 
most  original  mathematicians  of  antiquity,  Archimedes  and 
Apottonius  of  Perga.  They  rank  among  the  greatest  mathe- 
maticians of  all  time.  Only  a  small  part  of  their  discoveries 
can  be  described  here. 

Archimedes  (287  ?-212  B.C.)  was  born  at  Syracuse  in  Sicily. 
Cicero  tells  us  he  was  of  low  birth.  He  visited  Egypt  and, 
perhaps,  studied  in  Alexandria;  then  returned  to  his  native 
place,  where  he  made  himself  useful  to  his  admiring  friend  and 
patron,  King  Hieron,  by  applying  his  extraordinary  inventive 
powers  to  the  construction  of  war-engines,  by  which  he  inflicted 
great  loss  on  the  Romans,  who,  under  Marcellus,  were  besieging 
the  city.  That  by  the  use  of  mirrors  reflecting  the  sun's  rays 
he  set  on  fire  the  Roman  ships,  when  they  came  within  bow- 
shot of  the  walls,  is  probably  a  fiction.  Syracuse  was  taken  at 
length  by  the  Romans,  and  Archimedes  died  in  the  indiscrim- 
inate slaughter  which  followed.  The  story  goes  that,  at  the 
time,  he  was  studying  some  geometrical  diagram  drawn  in  the 
sand.  To  an  approaching  Roman  soldier  he  called  out,  "  Don't 
spoil  my  circles,"  but  the  soldier,  feeling  insulted,  killed  him. 
The  Roman  general  Marcellus,  who  admired  his  genius,  raised 
in  his  honour  a  tomb  bearing  the  figure  of  a  sphere  inscribed  in 
a  cylinder.  The  Sicilians  neglected  the  memory  of  Archimedes, 
for  when  Cicero  visited  Syracuse,  he  found  the  tomb  buried 
under  rubbish. 

While  admired  by  his  fellow-citizens  mainly  for  his  mechan- 
ical inventions,  he  himself  prized  more  highly  his  discoveries 
in  pure  science. 

Of  special  interest  to  us  is  his  book  on  the  Measurement  of 
the  Circle.1  He  proves  first  that  the  circular  area  is  equal  to 

1  A  recent  standard  edition  of  his  works  is  that  of  HEIBERG,  Leipzig, 
1880-81.  For  a  fuller  account  of  his  Measurement  of  the  Circle,  see 


76  A   HISTORY   OF   MATHEMATICS 

that  of  a  right  triangle  having  the  length  of  the  circumference 
for  its  base  and  the  radius  for  its  altitude.  To  find  this  base 
is  the  next  task.  He  first  finds  an  upper  limit  for  the  ratio  of 
the  circumference  to  the  diameter.  After  constructing  an  equi- 
lateral triangle  with  its  vertex  in  the  centre  of  the  circle  and 
its  base  tangent  to  the  circle,  he  bisects  the  angle  at  the  centre 
and  determines  the  ratio  of  the  base  to  the  altitude  of  one  of 
the  resulting  right  triangles,  taking  the  irrational  square  root 
a  little  too  small.  Next,  the  central  angle  of  this  right  triangle 
is  bisected  and  the  ratio  of  its  legs  determined.  Then  the  cen- 
tral angle  of  this  last  right  triangle  is  bisected  and  the  ratio  of 
its  legs  computed.  This  bisecting  and  computing  is  carried  on 
four  times,  the  irrational  square  roots  being  taken  every  time 
a  little  too  small.  The  ratio  of  the  last  two  legs  considered 
is  >  4673^ :  153.  But  the  shorter  of  the  legs  having  this  ratio 
is  the  side  of  a  regular  circumscribed  polygon.  This  leads  him 
to  the  conclusion  that  the  ratio  of  the  circumference  to  the 
diameter  is  <  3^-.  Next,  he  finds  a  lower  limit  by  inscribing 
regular  polygons  of  6,  12,  24,  48,  96  sides,  finding  for  each  suc- 
cessive polygon  its  perimeter.  In  this  way  he  arrives  at  the 
lower  limit  3f£.  Hence  the  final  result,  3|  >  IT  >  3-ff ,  an 
approximation  accurate  enough  for  most  purposes. 

Worth  noting  is  the  fact  that  while  approximations  to  TT  were 
made  before  this  time  by  the  Egyptians,  no  sign  of  such  com- 
putations occur  in  Euclid  and  his  Greek  predecessors.  Why 
this  strange  omission?  Perhaps  because  Greek  ideality  ex- 
cluded all  calculation  from  geometry,  lest  this  noble  science 
lose  its  rigour  and  be  degraded  to  the  level  of  geodesy  or  sur- 
veying. Aristotle  says  that  truths  pertaining  to  geometrical 
magnitudes  cannot  be  proved  by  anything  so  foreign  to  geom- 

CANTOR,  I.,  pp.  300-303,  316,  319  ;  LORIA,  II.,  pp.  126-132  ;  Gow,  pp.  233- 
237  ;  H.  WEISSENBORN,  Die  Berechnung  des  Kreis-  Umfanges  bei  Archi- 
medes und  Leonardo  Pisano,  Berlin,  1894. 


GREECE  77 

etry  as  arithmetic.  The  real  reason,  perhaps,  may  be  found 
in  the  contention  by  some  ancient  critics  that  it  is  not  evident 
that  a  straight  line  can  be  equal  in  length  to  a  curved  line ;  in 
particular  that  a  straight  line  exists  which  is  equal  in  length 
to  the  circumference.  This  involves  a  real  difficulty  in  geomet- 
rical reasoning.  Euclid  bases  the  equality  between  lines  or 
between  areas  on  congruence.  Now,  since  no  curved  line,  or 
even  a  part  of  a  curved  line,  can  be  made  to  exactly  coincide 
with  a  straight  line  or  even  a  part  of  a  straight  line,  no  com- 
parisons of  length  between  a  curved  line  and  a  straight  line 
can  be  made.  So,  in  Euclid,  we  nowhere  find  it  given  that  a 
curved  line  is  equal  to  a  straight  line.  The  method  employed 
in  Greek  geometry  truly  excludes  such  comparisons  ;  according 
to  Duhamel,  the  more  modern  idea  of  a  limit  is  needed  to  logi- 
cally establish  the  possibility  of  such  comparisons.1  On  Euclid- 
ean assumptions  it  cannot  even  be  proved  that  the  perimeter 
of  a  circumscribed  (inscribed)  polygon  is  greater  (smaller)  than 
the  circumference.  Some  writers  tacitly  resort  to  observation ; 
they  can  see  that  it  is  so. 

Archimedes  went  a  step  further  and  assumed  not  only  this, 
but,  trusting  to  his  intuitions,  tacitly  made  the  further  assump- 
tion that  a  straight  line  exists  which  equals  the  circumference 
in  length.  On  this  new  basis  he  made  a  valued  contribution  to 
geometry.  No  doubt  we  have  here  an  instance  of  the  usual 
course  in  scientific  progress.  Epoch-making  discoveries,  at 
their  birth,  are  not  usually  supported  on  every  side  by  unyield- 
ing logic ;  on  the  contrary,  intuitive  insight  guides  the  seeker 
over  difficult  places.  As  further  examples  of  this  we  instance 
the  discoveries  of  Newton  in  mathematics  and  of  Maxwell  in 
physics.  The  complete  chain  of  reasoning  by  which  the  truth 
of  a  discovery  is  established  is  usually  put  together  at  a  later 
period. 

1  G.  B.  HALSTED  in  Tr.  Texas  Academy  of  Science,  I.,  p.  96. 


7$  A   HISTORY   OF   MATHEMATICS 

Is  not  the  same  course  of  advancement  observable  in  the 
individual  mind  ?  We  first  arrive  at  truths  without  fully 
grasping  the  reasons  for  them.  Nor  is  it  always  best  that 
the  young  mind  should,  from  the  start,  make  the  effort  to 
grasp  them  all.  In  geometrical  teaching,  where  the  reasoning 
is  too  hard  to  be  mastered,  if  observation  can  conveniently 
assist,  accept  its  results.  A  student  cannot  wait  until  he  has 
mastered  limits  and  the  calculus,  before  accepting  the  truth 
that  the  circumference  is  greater  than  the  perimeter  of  an 
inscribed  polygon. 

Of  all  his  discoveries  Archimedes  prized  most  highly  those 
in  his  book  on  the  Sphere  and  Cylinder.  In  this  he  uses  the 
celebrated  statement,  "the  straight  line  is  the  shortest  path 
between  two  points,"  but  he  does  not  offer  this  as  a  formal 
definition  of  a  straight  line.1  Archimedes  proves  the  new 
theorems  that  the  surface  of  a  sphere  is  equal  to  four  times 
a  great  circle;  that  the  surface  of  a  segment  of  a  sphere  is 
equal  to  a  circle  whose  radius  is  the  straight  line  drawn  from 
the  vertex  of  the  segment  to  the  circumference  of  its  basal 
circle ;  that  the  volume  and  the  surface  of  a  sphere  are  %  of 
the  volume  and  surface,  respectively,  of  the  cylinder  circum- 
scribed about  the  sphere.  The  wish  of  Archimedes,  that  the 
figure  for  the  last  proposition  be  inscribed  upon  his  tomb,  was 
carried  out  by  the  Eoman  general  Marcellus. 

Archimedes  further  advanced  solid  geometry  by  adding  to 
the  five  "Platonic  figures,"  thirteen  semi-regular  solids,  each 
bounded  by  regular  polygons,  but  not  all  of  the  same  kind. 
To  elementary  geometry  belong  also  his  fifteen  Lemmas.2 

Of  the  "  Great  Geometer,"  Apollonius  of  Perga,  who  flour- 
ished about  forty  years  after  Archimedes,  and  investigated 
the  properties  of  the  conic  sections,  we  mention,  besides  his 

1  CANTOR,  I.,  298. 

2  Consult  Gow,  p.  232  ;  CANTOR,  I.,  p.  298. 


GREECE  79 

celebrated  work  on  Conies,  only  a  lost  work  on  Contacts, 
which  Vieta  and  others  attempted  to  restore  from  certain 
lemmas  given  by  Pappus.  It  contained  the  solution  of  the 
celebrated  "  Apollonian  Problem " :  Given  three  circles,  to 
find  a  fourth  which  shall  touch  the  three.  Even  in  modern 
times  this  problem  has  given  stimulus  toward  perfecting  geo- 
metric methods.1 

With  Euclid,  Archimedes,  and  Apollonius,  the  eras  of  Greek 
geometric  discovery  reach  their  culmination.  But  little  is 
known  of  the  history  of  geometry  from  the  time  of  Apollonius 
to  the  beginning  of  the  Christian  era.  In  this  interval  falls 
Zenodorus,  who  wrote  on  Figures  of  Equal  Periphery.  This 
book  is  lost,  but  fourteen  propositions  of  it  are  preserved  by 
Pappus  and  also  by  Theon.  Here  are  three  of  them:  "The 
circle  has  a  greater  area  than  any  polygon  of  equal  periphery," 
"Of  polygons  of  the  same  number  of  sides  and  of  equal 
periphery  the  regular  is  the  greatest,"  "  Of  all  solids  having 
surfaces  equal  in  area,  the  sphere  has  the  greatest  volume." 

Between  200  and  100  B.C.  lived  Hypsides,  the  supposed  author 
of  the  fourteenth  book  in  Euclid's  Elements.  His  treatise  on 
Risings  is  the  earliest  Greek  work  giving  the  division  of  the 
circle  into  360  degrees  after  the  manner  of  the  Babylonians. 

Hipparchus  of  Mcsea  in  Bithynia,  the  author  of  the  famous 
theory  of  epicycles  and  eccentrics,  is  the  greatest  astronomer 
of  antiquity.  Theon  of  Alexandria  tells  us  that  he  originated 
the  science  of  trigonometry  and  calculated  a  "  table  of  chords  " 
in  twelve  books  (not  extant).  Hipparchus  took  astronomical 
observations  between  161  and  126  B.C. 

A  writer  whose  tone  is  very  different  from  that  of  the  great 
writers  of  the  First  Alexandrian  School  was  Heron  of  Alexan- 


1  Consult  E.  SCHILKE,  Die  Losungen  und  Erweiterungen  des  Apolloni- 
schen  Beriihrungsproblems,  Berlin,  1880. 


80  A   HISTORY   OF   MATHEMATICS 

dria,  also  called  Heron  the  Elder.1  As  he  was  a  practical 
surveyor,  it  is  not  surprising  to  find  little  resemblance  between 
his  writings  and  those  of  Euclid  or  Apollonius. 

Heron  was  a  pupil  of  Ctesibius,  who  was  celebrated  for  his 
mechanical  inventions,  such  as  the  hydraulic  organ,  water-clock, 
and  catapult.  It  is  believed  by  some  that  Heron  was  a  son 
of  Ctesibius.  Heron's  invention  of  the  eolipile  and  a  curious 
mechanism,  known  as  "  Heron's  Fountain,"  display  talent  of 
the  same  order  as  that  of  his  master.  Great  uncertainty  exists 
regarding  his  writings.  Most  authorities  believe  him  to  be 
the  author  of  a  work,  entitled  Dioptra,  of  which  three  quite 
dissimilar  manuscripts  are  extant.  Marie2  thinks  that  the 
Dioptra  is  the  work  of  a  writer  of  the  seventh  or  eighth  cen- 
tury A.D.,  called  Heron  the  Younger.  But  we  have  no  reliable 
evidence  that  a  second  mathematician  by  the  name  of  Heron 
really  existed.3  A  reason  adduced  by  Marie  for  the  later 
origin  of  the  Dioptra  is  the  fact  that  it  is  the  first  work  which 
contains  the  important  formula  for  the  area  of  a  triangle,  ex- 
pressed in  terms  of  the  three  sides.  Now,  not  a  single  Greek 
writer  cites  this  formula ;  hence  he  thinks  it  improbable  that 
the  Dioptra  was  written  as  early  as  the  time  of  Heron  the 
Elder.  This  argument  is  not  convincing,  for  the  reason  that 
only  a  small  part  of  Greek  mathematical  literature  of  this 
period  has  been  preserved.  The  formula,  sometimes  called 
"  Heronic  formula,"  expresses  the  triangular  area  as  follows : 


-j-  c    a  -f-  b  —  c     a  -f-  c  —  b     b  +  c  —  a 
~~2~~  ~~2~~          ~~2~~ 

where  a,  6,  c  are  the  sides.    The  proof  given,  though  laborious, 

1  According  to  CANTOR,  I.,  363,  he  flourished  about  100  B.C.  ;  according 
to  MARIE,  I.,  177,  about  155  B.C. 

2  MARIE,  I.,  180. 

8  CANTOR,  I.,  367. 


GKEECE  81 

is  exceedingly  ingenious  and  discloses  no  little  mathematical 
ability.1 

"  Dioptra,"  says  Venturi,  "  were  instruments  resembling  our 
modern  theodolites.  The  instrument  consisted  of  a  rod  four 
yards  long  with  little  plates  at  the  ends  for  aiming.  This 
rested  upon  a  circular  disc.  The  rod  could  be  moved  horizon- 
tally and  also  vertically.  By  turning  the  rod  around  until 
stopped  by  two  suitably  located  pins  on  the  circular  disc,  the 
surveyor  could  work  off  a  line  perpendicular  to  a  given  direc- 
tion. The  level  and  plumb-line  were  used." 2  Heron  explains 
with  aid  of  these  instruments  and  of  geometry  a  large  num- 
ber of  problems,  such  as  to  find  the  distance  between  two 
points  of  which  one  only  is  accessible,  or  between  two  points 
which  are  visible,  but  both  inaccessible ;  from  a  given  point  to 
run  a  perpendicular  to  a  line  which  cannot  be  approached; 
to  find  the  difference  of  level  between  two  points ;  to  measure 
the  area  of  a  field  without  entering  it. 

The  Dioptra  discloses  considerable  mathematical  ability  and 
familiarity  with  the  writings  of  the  authors  of  the  classical 
period.  Heron  had  read  Hipparchus  and  he  wrote  a  commen- 
tary on  Euclid.3  Nevertheless  the  character  of  his  geometry 
is  not  Grecian,  but  Egyptian.  Usually  he  gives  directions  and 
rules,  without  proofs.  He  gives  formulae  for  computing  the 
area  of  a  regular  polygon  from  the  square  of  one  of  its  sides ; 
this  implies  a  knowledge  of  trigonometry.  Some  of  Heron's 
formulae  point  to  an  old  Egyptian  origin.  Thus,  besides  the 

formula  for  a  triangular  area,  given  above,  he  gives 
which  bears  a  striking  likeness  to  the  formula 

1  For  the  proof,  see  Dioptra  (Ed.  HULTSCH),  pp.  235-23?  ;  CANTOR,  I., 
375  ;  Gow,  p.  281. 

2  CANTOR,  I.,  382.  8  See  TANNERY,  pp.  165-181. 

G 


82  A    HISTORY   OF   MATHEMATICS 

for  finding  the  area  of  a  quadrangle,  found  in  the  Edf u  inscrip- 
tions. There  are  points  of  resemblance  between  Heron  and 
Ahmes.  Thus,  Ahmes  used  unit-fractions  exclusively ;  Heron 
used  them  oftener  than  other  fractions.  That  the  arithmetical 
theories  of  Ahmes  were  not  forgotten  at  this  time  is  also 
demonstrated  by  the  Akhmim  papyrus,  which,  though  the 
oldest  extant  text-book  on  practical  Greek  arithmetic,  was 
probably  written  after  Heron's  time.  Like  Ahmes  and  the 
priests  at  Edfu,  Heron  divides  complicated  figures  into  simpler 
ones  by  drawing  auxiliary  lines;  like  them  he  displays  par- 
ticular fondness  for  the  isosceles  trapezoid. 

The  writings  of  Heron  satisfied  a  practical  want,  and  for 
that  reason  were  widely  read.  We  find  traces  of  them  in 
Rome,  in  the  Occident  during  the  Middle  Ages,  and  even  in 
India. 

VI.  The  Second  Alexandrian  School.  —  With  the  absorption 
of  Egypt  into  the  Roman  Empire  and  with  the  spread  of 
Christianity,  Alexandria  became  a  great  commercial  as  well  as 
intellectual  centre.  Traders  of  all  nations  met  in  her  crowded 
streets,  while  in  her  magnificent  library,  museums,  and  lecture- 
rooms,  scholars  from  the  East  mingled  with  those  of  the  West. 
Greek  thinkers  began  to  study  the  Oriental  literature  and 
philosophy.  The  resulting  fusion  of  Greek  and  Oriental 
philosophy  led  to  Neo-Pythagoreanism  and  Neo-Platonism. 
The  study  of  Platonism  and  Pythagorean  mysticism  led  to  a 
revival  of  the  theory  of  numbers.  This  subject  became  again 
a  favourite  study,  though  geometry  still  held  an  important 
place.  This  second  Alexandrian  school,  beginning  with  the 
Christian  era,  was  made  famous  by  the  names  of  Diophan- 
tus,  Claudius  Ptolemaeus,  Pappus,  Theon  of  Smyrna,  Theon 
of  Alexandria,  lamblichus,  Porphyrius,  Serenus  of  Antinoeia, 
Menelaus,  and  others. 

Menelaus  of  Alexandria  lived  about  98  A.D.,  as  appears  from 


GREECE  83 

two  astronomical  observations  taken  by  him  and  recorded  in 
the  Almagest.1  Valuable  contributions  were  made  by  him  to 
spherical  geometry,  in  his  work,  Sphcerica,  which  is  extant  in 
Hebrew  and  Arabic,  but  lost  in  the  original  Greek.  He  gives 
the  theorems  on  the  congruence  of  spherical  triangles  and  de- 
scribes their  properties,  in  much  the  same  way  as  Euclid  treats 
plane  triangles.  He  gives  the  theorems  that  the  sum  of  the 
three  sides  of  a  spherical  triangle  is  less  than  a  great  circle, 
and  that  the  sum  of  three  angles  exceeds  two  right  angles. 
Celebrated  are  two  theorems  of  his  on  plane  and  spherical  tri- 
angles. The  one  on  plane  triangles  is  that  "  if  the  three  sides 
be  cut  by  a  straight  line,  the  product  of  the  one  set  of  three 
segments  which  have  no  common  extremity  is  equal  to  the  prod- 
uct of  the  other  three."  The  illustrious  Lazare  Carnot  makes 
this  proposition,  known  as  the  "  lemma  of  Menelaus,"  the  base 
of  his  theory  of  transversals.2  The  corresponding  theorem  for 
spherical  triangles,  the  so-called  "rule  of  six  quantities,"  is 
obtained  from  the  above  by  reading  "  chords  of  three  segments 
doubled,"  in  place  of  "  three  segments." 

Another  fundamental  theorem  in  modern  geometry  (in  the 
theory  of  harmonics)  is  the  following  ascribed  to  Serenus  of 
Antinoeia :  If  from  D  we  draw  DF,  cutting  the  triangle  ABC, 
and  choose  H  on  it,  so  that  DE :  DF  =  RE :  HF,  and  if  we 
draw  the  line  AH,  then  every  transversal  through  D,  such  as 


1  CANTOR,  L,  412. 

2  For  the  history  of  the  theorem  see  M.  CHASLES,  Geschichte  der  Geom- 
etrie.     Aus  dem  Franzosischen  tibertragen  durch  DR.  L.  A.  SOHNCKE, 
Halle,  1839,  Note  VI.,  pp.  295-299.      Hereafter  we  quote  this  work  as 
CHASLES.     A  recent  French  edition  of  this  important  work  is  now  easily 
obtainable.     Chasles  points  out  that  the  "  lemma  of  Menelaus  "  was  well 
known  during  the   sixteenth  and  seventeenth  centuries,  but  from  that 
time,  for  over  a  century,  it  was  fruitless  and  hardly  known,  until  finally 
Carnot  began  his  researches.     Carnot,  as  well  as  Ceva,  rediscovered  the 
theorem. 


84  A   HISTOBY   OF   MATHEMATICS 

DG,  will  be  divided  by  AH  so  that  DK:  DG  =  JK:  JG.    As 

Antinoeia  (or  Antinoupolis)  in  Egypt  was  founded  by  Emperor 

Hadrian  in  122  A.D.,  we  have 
an  upper  limit  for  the  date 
of  Serenus.1  The  fact  that 
he  is  quoted  by  a  writer  of 
the  fifth  or  sixth  century  sup- 
plies us  with  a  lower  limit. 
A  central  position  in  the  history  of  ancient  astronomy  is 
occupied  by  Claudius  Ptolemceus.  Nothing  is  known  of  his 
personal  history  except  that  he  was  a  native  of  Egypt  and 
flourished  in  Alexandria  in  139  A.D.  The  chief  of  his  works 
are  the  Syntaxis  Mathematica  (or  the  Almagest,  as  the  Arabs 
called  it)  and  the  Geographica.  This  is  not  the  place  to  de- 
scribe the  "Ptolemaic  System";  we  mention  Ptolemseus  because 
of  the  geometry  and  especially  the  trigonometry  contained  in 
the  Almagest.  He  divides  the  circle  into  360  degrees;  the 
diameter  into  120  divisions,  each  of  these  into  60  parts,  which 
are  again  divided  into  60  smaller  parts.  In  Latin,  these  parts 
were  called  partes  minutce  primce  and  partes  minutce  secundce.2 
Hence  our  names  "minutes"  and  "seconds."  Thus  the  Baby- 
lonian sexagesimal  system,  known  to  Geminus  and  Hipparchus, 
had  by  this  time  taken  firm  root  among  the  Greeks  in  Egypt. 
The  foundation  of  trigonometry  had  been  laid  by  the  illustrious 
Hipparchus.  Ptolemaeus  imparted  to  it  a  remarkably  perfect 
form.  He  calculated  a  table  of  chords  by  a  method  which 
seems  original  with  him.  After  proving  the  proposition,  now 
appended  to  Euclid,  VI.  (D),  that  "  the  rectangle  contained  by 
the  diagonals  of  a  quadrilateral  figure  inscribed  in  a  circle  is 
equal  to  both  the  rectangles  contained  by  its  opposite  sides," 
he  shows  how  to  find  from  the  chords  of  two  arcs  the  chords  of 

1  J.  L.  HEIBERG  in  Bibliotheca  Mathematica,  1894,  p.  97. 

2  CANTOR,  I.,  p.  416. 


GREECE  85 

their  sum  and  difference,  and  from  the  chord  of  any  arc  that 
of  its  half.  These  theorems,  to  which  he  gives  pretty  proofs,1, 
are  applied  to  the  calculation  of  chords.  It  goes  without  say- 
ing that  the  nomenclature  and  notation  (so  far  as  he  had  any) 
were  entirely  different  from  those  of  modern  trigonometry.  In 
place  of  our  "  sine,"  he  considers  the  "  chord  of  double  the  arc." 
Thus,  in  his  table,  the  chord  21  •  21  •  12  is  given  for  the  arc 
20°  30'.  Eeducing  the  sexagesimals  to  decimals,  we  get  for  the 
chord  .35588.  Its  half,  .17794,  is  seen  to  be  the  sine  of  10°  15', 
or  the  half  of  20°  30'. 

More  complete  than  in  plane  trigonometry  is  the  theoretical 
exposition  of  propositions  in  spherical  trigonometry.2  Ptolemy 
starts  out  with  the  theorems  of  Menelaus.  The  fact  that 
trigonometry  was  cultivated  not  for  its  own  sake,  but  to  aid 
astronomical  inquiry,  explains  the  rather  startling  fact  that 
spherical  trigonometry  came  to  exist  in  a  developed  state 
earlier  than  plane  trigonometry. 

Of  particular  interest  to  us  is  a  proof  which,  according  to 
Proclus,  was  given  by  Ptolemy  to  Euclid's  parallel-postulate. 
The  critical  part  of  the  proof  is  as  follows:   If  the  straight 
lines  are  parallel,  the  interior  angles  [on  the  same  side  of  the 
transversal]  are  necessarily  equal  to 
two  right  angles.     For  «£,  yrj  are  not                         r 
less  parallel  than  £/?,  ^8  and,  there-    ./^ -P- -\ 

^\.  I  ^^K 

fore,  whatever  the  sum  of  the  angles        y -h f 

/3£rj,  £ryS,  whether  greater  or  less  than  / 

two  right  angles,  such  also  must  be 

the  sum  of  the  angles  a£rj,  £777.      But  the  sum  of  the  four 

cannot  be  more  than  four  right  angles,  because  they  are  two 

pairs  of  adjacent  angles.3     The  untenable  point  in  this  proof 

1  Consult  CANTOR,  I.,  417,  for  the  proofs. 

2  Consult  CANTOR,  I.,  p.  420  ;  Gow,  p.  297. 
8  Gow,  p.  301. 


86  A   HISTORY   OF   MATHEMATICS 

is  the  assertion  that,  in  case  of  parallelism,  the  sum  of  the 
interior  angles  on  one  side  of  the  transversal  must  be  the  same 
as  their  sum  on  the  other  side  of  the  transversal.  Ptolemy 
appears  to  be  the  first  of  a  long  line  of  geometers  who  dur- 
ing eighteen  centuries  vainly  attempted  to  prove  the  parallel- 
postulate,  until  finally  the  genius  of  Lobatchewsky  and  Bolyai 
dispelled  the  haze  which  obstructed  the  vision  of  mathemati- 
cians and  led  them  to  see  with  unmistakable  clearness  the 
reason  why  such  proofs  have  been  and  always  will  be  futile. 

For  150  years  after  Ptolemy,  there  appeared  no  geometer 
of  note.  An  unimportant  work,  entitled  Cestes,  by  Sextus 
Julius  African-us,  applies  geometry  to  the  art  of  war.  Pappus 
(about  300  or  370,  A.D.)  was  the  last  great  mathematician  of 
the  Alexandrian  school.  Though  inferior  in  genius  to  Archi- 
medes, Apollonius,  and  Euclid,  who  wrote  five  centuries  earlier, 
Pappus  towers  above  his  contemporaries  as  does  a  lofty  peak 
above  the  surrounding  plains.  Of  his  several  works  the  Mathe- 
matical Collections  is  the  only  one  extant.  It  was  in  eight 
books  ;  the  first  and  parts  of  the  second  are  now  missing.  The 
object  of  this  treatise  appears  to  have  been  to  supply  geometers 
with  a  succinct  analysis  of  the  most  difficult  mathematical 
works  and  to  facilitate  the  study  of  these  by  explanatory 
lemmas.  It  is  invaluable  to  us  on  account  of  the  rich 
information  it  gives  on  various  treatises  by  the  foremost 
Greek  mathematicians,  which  are  now  lost.  Scholars  of  the 
last  century  considered  it  possible  to  restore  lost  works  from 
the  resume  given  by  Pappus  alone.  Some  of  the  theorems 
are  doubtless  original  with  Pappus,  but  here  it  is  difficult  to 
speak  with  certainty,  for  in  three  instances  Pappus  copied 
theorems  without  crediting  them  to  the  authors,  and  he  may 
have  done  the  same  in  other  cases  where  we  have  no  means  of 
ascertaining  the  real  discoverer. 

Of  elementary  propositions  of  special  interest  and  probably 


GREECE  87 

his  own,  we  mention  the  following :  (1)  The  centre  of  inertia 
(gravity)  of  a  triangle  is  that  of  another  triangle  whose  vertices 
lie  upon  the  sides  of  the  first  and  divide  its  three  sides  in 
the  same  ratio ;  (2)  solution  of  the  problem  to  draw  through 
three  points  lying  in  the  same  straight  line,  three  straight 
lines  which  shall  form  a  triangle  inscribed  in  a  given  circle ; l 
(3)  he  propounded  the  theory  of  involution  of  points ;  (4)  the 
line  connecting  the  opposite  extremities  of  parallel  diameters 
of  two  externally  tangent  circles  passes  through  the  point  of 
contact  (a  theorem  suggesting  the  consideration  of  centres 
of  similitude  of  two  circles) ;  (5)  solution  of  the  problem,  to 
find  a  parallelogram  whose  sides  are  in  a  fixed  ratio  to  those  of 
a  given  parallelogram,  while  the  areas  of  the  two  are  in  another 
fixed  ratio.  This  resembles  somewhat  an  indeterminate  prob- 
lem given  by  Heron,  to  construct  two  rectangles  in  which  the 
sums  of  their  sides,  as  well  as  their  areas,  are  in  given  ratios.2 

It  remains  for  us  to  name  a  few  more  mathematicians. 
Theon  of  Alexandria  brought  out  an  edition  of  Euclid's  Ele- 
ments, with  notes ;  his  commentary  on  the  Almagest  is  valuable 
on  account  of  the  historical  notes  and  the  specimens  of  Greek 
arithmetic  found  therein.  Theon's  daughter  Hypatia,  a  woman 
renowned  for  her  beauty  and  modesty,  was  the  last  Alexandrian 
teacher  of  reputation.  Her  notes  on  Diophantus  and  Apollo- 
nius  have  been  lost.  Her  tragic  death  in  415  A.D.  is  vividly 
described  in  Kingsley's  Hypctfia.* 

From  now  on  Christian  theology  absorbed  men's  thoughts. 

1  "The  problem,  generalized  by  placing  the  points  anywhere,  has 
become  celebrated,  partly  by  its  difficulty,  partly  by  the  names  of  geom- 
eters who  solved  it,  and  especially  by  the  solution,  as  general  as  it  was 
simple,  given  by  a  boy  of  16,  Ottaiano  of  Naples."  CHASLES,  p.  41 ; 
Chasles  gives  a  history  of  the  problem  in  note  XL  2  CANTOR,  I.,  454. 

3  The  reader  may  be  interested  in  an  article  by  G.  VALENTIN,  "  Die 
Frauen  in  den  exakten  Wissenschaften,"  Bibliotheca  Mathematica,  1895, 
pp.  65-76. 


88  A    HISTORY    OF   MATHEMATICS 

1 

In  Alexandria  Paganism  disappeared  and  with  it  Pagan  learn- 
ing. The  Neo-Platonic  school  at  Athens  struggled  a  century 
longer.  Proclus,  Isidorus,  and  others  endeavoured  to  keep  up 
"the  golden  chain  of  Platonic  succession."  Proclus  wrote  a 
commentary  on  Euclid ;  that  on  the  first  book  is  extant  and  is 
of  great  historical  value.  A  pupil  of  Isidorus,  Damascius  of 
Damascus  (about  510  A.D.)  is  believed  by  some  to  be  the 
author  of  Book  XV.  of  Euclid's  Elements. 

The  geometers  of  the  last  500  years,  with  the  possible 
exception  of  Pappus,  lack  creative  power ;  they  are  commenta- 
tors rather  than  originators. 

The  salient  features  of  Greek  geometry  are : 

(1)  A  wonderful    clearness    and    defmiteness    of   concepts, 
and  an  exceptional   logical  rigour  of  conclusions.     We  have 
encountered  occasional  flaws  in  reasoning ;  but  when  we  com- 
pare Greek  geometry  in  its  most  complete  form  with  the  best 
that  the  Babylonians,   Egyptians,   Romans,  Hindus,   or   the 
geometers  of  the  Middle   Ages  have  brought  forth,  then  it 
must  be  admitted  that  not  only  in  rigour  of  presentation,  but 
also  in  fertility  of  invention,  the  geometric  mind  of  the  Greek 
towers  above  all  others  in  solitary  grandeur. 

(2)  A  complete  absence  of  general  principles  and  methods. 
For  example,   the   Greeks   possessed  no   general  method  of 
drawing  tangents.     In  the  demonstration  of  a  theorem,  there 
were,  for  the  ancient   geometers,  as   many  different  cases  re- 
quiring separate  proof  as  there  were  different  positions  for  the 
lines.1     "It  is   one  of  the   greatest  advantages  of  the  more 
modern  geometry  over  the  ancient  that  through  the  considera- 
tion of  positive  and   negative    quantities  it  embraces  in  one 
expression  the  several  cases  which  a  theorem  can  present  in 
respect  to  the  various  relative  positions  of  the  separate  parts 

1  Take,  for  example,  EUCLID,  III.,  35. 


ROME  89 

of  the  figure.  Thus  to-day  the  nine  main  problems  and  the 
numerous  special  cases,  which  are  the  subject-matter  of  83 
theorems  in  the  two  books  de  sectione  determinata  (of  Pappus), 
constitute  only  one  problem  which  can  be  solved  by  one  single 
equation."1  "If  we  compare  a  mathematical  problem  with 
a  huge  rock,  into  the  interior  of  which  we  desire  to  penetrate, 
then  the  work  of  the  Greek  mathematicians  appears  to  us 
like  that  of  a  vigorous  stonecutter  who,  with  chisel  and  ham- 
mer, begins  with  indefatigable  perseverance,  from  without,  to 
crumble  the  rock  slowly  into  fragments ;  the  modern  mathe- 
matician appears  like  an  excellent  miner,  who  first  bores 
through  the  rock  some  few  passages,  from  which  he  then 
bursts  it  into  pieces  with  one  powerful  blast,  and  brings  to 
light  the  treasures  within.'7  2 


EOME 

Although  the  Romans  excelled  in  the  science  of  government 
and  war,  in  philosophy,  poetry,  and  art  they  were  mere  imita- 
tors. In  mathematics  they  did  not  even  rise  to  the  desire  for 
imitation.  If  we  except  the  period  of  decadence,  during 
which  the  reading  of  Euclid  began,  we  can  say  that  the  classical 
Greek  writers  on  geometry  were  wholly  unknown  in  Rome. 
A  science  of  geometry  with  definitions,  postulates,  axioms, 
rigorous  proofs,  did  not  exist  there.  A  practical  geometry, 
like  the  old  Egyptian,  with  empirical  rules  applicable  in  sur- 
veying, stood  in  place  of  the  Greek  science.  Practical  treatises 
prepared  by  Roman  surveyors,  called  agrimensores  or  gromatici, 
have  come  down  to  us.  "  As  regards  the  geometrical  part  of 

1  CHASLES,  p.  39. 

2  HERMANN  HANKEL,  Die  Entwickelung  der  Mathematik  in  den  letzten 
Jahrhunderten,  Tiibmgen,  1884,  p.  9. 


90  A   HISTORY    OF   MATHEMATICS 

these  pandects,  which  treat  exhaustively  also  of  the  juristic 
and  purely  technical  side  of  the  art,  it  is  difficult  to  say 
whether  the  crudeness  of  presentation,  or  the  paucity  and 
faultiness  of  the  contents  more  strongly  repels  the  reader. 
The  presentation  is  beneath  the  notice  of  criticism,  the  termi- 
nology vacillating;  of  definitions  and  axioms  or  proofs  of  the 
prescribed  rules  there  is  no  mention.  The  rules  are  not  formu- 
lated; the  reader  is  left  to  abstract  them  from  numerical 
examples  obscurely  and  inaccurately  described.  The  total 
impression  is  as  though  the  Roman  gromatici  were  thousands 
of  years  older  than  Greek  geometry,  and  as  though  the  deluge 
were  lying  between  the  two."1  Some  of  their  rules  were 
probably  inherited  from  the  Etruscans,  but  others  are  identi- 
cal with  those  of  Heron.  Among  the  latter  is  that  for  finding 
the  area  of  a  triangle  from  its  sides  [the  "  Heronic  formula  "] 
and  the  approximate  formula,  Jf  a2,  for  the  area  of  equilateral 
triangles  (a  being  one  of  the  sides).  But  the  latter  area  was 
also  calculated  by  the  formulas  ^-(a2  +  a)  and  1  a2,  the  first  of 
which  was  unknown  to  Heron.  Probably  -|-a2  was  derived 
from  an  Egyptian  formula.  The  more  elegant  and  refined 
methods  of  Heron  were  unknown  to  the  Romans.  The  grom- 
atici considered  it  sometimes  sufficiently  accurate  to  determine 
the  areas  of  cities  irregularly  laid  out,  simply  by  measuring 
their  circumferences.2  Egyptian  geometry,  or  as  much  of  it 
as  the  Romans  thought  they  could  use,  was  imported  at  the 
time  of  Julius  Csesar,  who  ordered  a  survey  of  the  whole 
empire  to  secure  an  equitable  mode  of  taxation.  From  early 
times  it  was  the  Roman  practice  to  divide  land  into  rectan- 
gular and  rectilinear  parts.  Walls  and  streets  were  parallel, 
enclosing  squares  of  prescribed  dimensions.  This  practice 

1  HANKEL,  pp.  295,  296.     For  a  detailed  account  of  the  agrimensores, 
consult  CANTOR,  I.,  pp.  521-538. 

2  HANKEL,  p.  297. 


ROME  91 

simplified  matters  immensely  and  greatly  reduced  the  neces- 
sary amount  of  geometrical  knowledge.  Approximate  formulae 
answered  all  ordinary  demands  of  precision. 

Caesar  reformed  the  calendar,  and  for  this  undertaking  drew 
from  Egyptian  learning.  The  Alexandrian  astronomer  Sosi- 
genes  was  enlisted  for  this  task.  Among  Roman  names  identi- 
fied with  geometry  or  surveying,  are  the  following:  Marcus 
Terentius  Varro  (about  116-27  B.C.),  Sextus  Julius  Frontinus  (in 
70  A.D.  praetor  in  Rome),  Martianus  Mineus  Felix  Capella 
(born  at  Carthage  in  the  early  part  of  the  fifth  century), 
Magnus  Aurelius  Cassiodorius  (born  about  475  A.D.).  Vastly 
superior  to  any  of  these  were  the  Greek  geometers  belonging 
to  the  period  of  decadence  of  Greek  learning. 

It  is  a  remarkable  fact  that  the  period  of  political  humilia- 
tion, marked  by  the  fall  of  the  Western  Roman  Empire  and 
the  ascendancy  of  the  Ostrogoths,  is  the  period  during  which 
the  study  of  Greek  science  began  in  Italy.  The  compila- 
tions made  at  this  time  are  deficient,  yet  interesting  from  the 
fact  that,  down  to  the  twelfth  century,  they  were  the  only 
sources  of  mathematical  knowledge  in  the  Occident.  Fore- 
most among  these  writers  is  Anicius  Manlius  Severinus 
Boethius  (480?-524).  At  first  a  favourite  of  King  Theod- 
oric,  he  was  later  charged  with  treason,  imprisoned,  and 
finally  decapitated.  While  in  prison  he  wrote  On  the  Con- 
solations of  Philosophy.  Boethius  wrote  an  Institutio  Arith- 
metics (essentially  a  translation  of  the  arithmetic  of  Mco- 
machus)  and  a  Geometry.  The  first  book  of  his  Geometry  is 
an  extract  from  the  first  three  books  of  Euclid's  Elements, 
with  the  proofs  omitted.  It  appears  that  Boethius  and  a  num- 
ber of  other  writers  after  him  were  somehow  led  to  the  belief 
that  the  theorems  alone  belonged  to  Euclid,  while  the  proofs 
were  interpolated  by  Theon ;  hence  the  strange  omission  of  all 
demonstration.  The  second  book  in  the  Geometry  of  Boethius 


92  A  HISTORY   OF  MATHEMATICS 

consists  of  an  abstract  of  the  practical  geometry  of  Frontmus, 
the  most  accomplished  of  the  gromatici. 

Notice  that,  imitating  Nicomachus,  Boethius  divides  the 
mathematical  sciences  into  four  sections,  Arithmetic,  Music, 
Geometry,  Astronomy.  He  first  designated  them  by  the  word 
quadruvium  (four  path-ways).  This  term  was  used  extensively 
during  the  Middle  Ages.  Cassiodorius  used  a  similar  figure, 
the  four  gates  of  science.  Isidorus  of  Carthage  (born  570),  in 
his  Origines,  groups  all  sciences  as  seven,  the  four  embraced 
by  the  quadruvium  and  three  (Grammar,  Ehetoric,  Logic) 
which  constitute  the  trivium  (three  path-ways). 


MIDDLE  AGES 


ARITHMETIC   AND  ALGEBRA 
Hindus 

SOON  after  the  decadence  of  Greek  mathematical  research, 
another  Aryan  race,  the  Hindus,  began  to  display  brilliant 
mathematical  power.  Not  in  the  field  of  geometry,  but  of 
arithmetic  and  algebra,  they  achieved  glory.  In  geometry 
they  were  even  weaker  than  were  the  Greeks  in  algebra.  The 
subject  of  indeterminate  analysis  (not  within  the  scope  of  this 
history)  was  conspicuously  advanced  by  them,  but  on  this 
point  they  exerted  no  influence  on  European  investigators, 
for  the  reason  that  their  researches  did  not  become  known 
in  the  Occident  until  the  nineteenth  century. 

India  had  no  professed  mathematicians ;  the  writers  we  are 
about  to  discuss  considered  themselves  astronomers.  To  them, 
mathematics  was  merely  a  handmaiden  to  astronomy.  In 
view  of  this  it  is  curious  to  observe  that  the  auxiliary 
science  is  after  all  the  only  one  in  which  they  won  real  dis- 
tinction, while  in  their  pet  pursuit  of  astronomy  they  displayed 
an  inaptitude  to  observe,  to  collect  facts,  and  to  make  inductive 
investigations. 

It  is  an  unpleasant  feature  about  the  Hindu  mathematical 
treatises  handed  down  to  us  that  rules  and  results  are  expressed 
in  verse  and  clothed  in  obscure  mystic  language.  To  him  who 

93 


94  A  HISTORY   OF   MATHEMATICS 

already  understands  the  subject  such  verses  may  aid  the 
memory,  but  to  the  uninitiated  they  are  often  unintelligible. 
Usually  proofs  are  not  preserved,  though  Hindu  mathemati- 
cians doubtless  reasoned  out  all  or  most  of  their  discoveries. 

It  is  certain  that  portions  of  Hindu  mathematics  are  of 
Greek 'origin.  An  interesting  but  difficult  task  is  the  tracing 
of  the  relation  between  Hindu  and  Greek  thought.  After 
Egypt  had  become  a  Roman  province,  extensive  commercial 
relations  sprang  up  with  Alexandria.  Doubtless  there  was 
considerable  interchange  of  philosophic  and  scientific  know- 
ledge. The  Hindus  were  indebted  to  Heron,  Diophantus  and 
Ptolemy.  They  were  borrowers  also  from  the  Chinese. 

At  present  we  know  very  little  of  the  growth  of  Hindu 
mathematics.  The  few  works  handed  down  to  us  exhibit  the 
science  in  its  complete  state  only.  The  dates  of  all  important 
works  but  the  first  are  well  fixed.  In  1881  there  was  found 
at  Bakhshali,  in  northwest  India,  buried  in  the  earth,  an 
anonymous  arithmetic,  supposed,  from  the  peculiarities  of  its 
verses,  to  date  from  the  third  or  fourth  century  after  Christ. 
The  document  that  was  found  is  of  birch  bark  and  is  an  in- 
complete copy,  prepared  probably  about  the  eighth  century, 
of  an  older  manuscript.1 

The  earliest  Hindu  astronomer  known  to  us  is  Aryabhatta, 
born  in  476  A.D.,  at  Pataliputra,  on  the  upper  Ganges.  He 
is  the  author  of  the  celebrated  work,  entitled  Aryabhattiyam, 
the  third  chapter  of  which  is  given  to  mathematics.2  About 
one  hundred  years  later  flourished  Brahmagupta,  born  598,  who 
in  628  wrote  his  Brakma-sphuta-siddhanta  ("  The  revised  system 
of  Brahma"),  of  which  the  twelfth  and  eighteenth  chapters 

1  CANTOR,   I.,  pp.  598,  613-615.     See  also  The  Bakhshali  Manuscript, 
edited  by  RUDOLF  HOERNLE  in  the  Indian  Antiquary,  XVII.,  33-48  and 
275-279,  Bombay,  1888. 

2  Translated  by  L.  RODET  in  Journal  Asiatique,  1879,  s6rie  7,  T.  XIIL 


HINDUS  95 

belong  to  mathematics.1  Important  is  an  elementary  work  by 
Mahavira  (9th  century  ?)  ;  then  come,  Cridhara,  who  wrote  a 
Ganitarsara  ("  Quintessence  of  Calculation  "),  and  Padmanabha, 
the  author  of  an  algebra.  The  science  seems  to  have  made 
but  little  progress  since  Brahmagupta,  for  a  work  entitled 
Siddhantaciromani  ("Diadem  of  an  Astronomical  System"), 
written  by  Bhaskara  Acarya  in  1150,  stands  little  higher  than 
Brahmagupta' s  Siddhanta,  written  over  five  hundred  years 
earlier.  The  two  most  important  mathematical  chapters  in 
Bhaskara' s  work  are  the  Lilavati  ("the  beautiful,"  i.e.  the 
noble  science),  and  Viga-ganita  ("root-extraction"),  devoted  to 
arithmetic  and  algebra.  From  this  time  the  spirit  of  research 
was  extinguished  and  no  great  names  appear. 

Elsewhere  we  have  spoken  of  the  Hindu's  clever  use  of  the 
principle  of  local  value  and  of  the  zero  in  arithmetical  nota- 
tion. We  now  give  an  account  of  Hindu  methods  of  computa- 
tion, which  were  elaborated  in  India  to  a  perfection  undreamed 
of  by  earlier  nations.  The  information  handed  down  to  our 
time  is  derived  partly  from  the  Hindu  works,  but  mainly  from 
an  arithmetic  written  by  a  Greek  monk,  Maximus  Planudes, 
who  lived  in  the  first  half  of  the  fourteenth  century,  and  who 
avowedly  used  Hindu  sources. 

To  understand  the  reason  why  certain  modes  of  computation 
were  adopted,  we  must  bear  in  mind  the  instruments  at 'the 
disposal  of  the  Hindus  in  their  written  calculations.  They 

1  Translated  into  English  by  H.  T.  COLEBROOKE,  London,  1817.  This 
celebrated  Sanskrit  scholar  translated  also  the  mathematical  chapters  of 
the  Siddhantaciromani  of  Bhaskara.  Colebrooke  held  a  judicial  office  in 
India,  and  about  the  year  1794  entered  upon  the  study  of  Sanskrit,  that 
he  might  read  Hindu  law-books.  From  youth  fond  of  mathematics,  he 
also  began  to  study  Hindu  astronomy  and  mathematics ;  and,  finally,  by 
his  translations,  demonstrated  to  Europe  the  fact  that  the  Hindus  in 
previous  centuries  had  made  remarkable  discoveries,  some  of  which  had 
been  wrongly  ascribed  to  the  Arabs. 


96  A   HISTORY   OF   MATHEMATICS 

wrote  "  with  a  cane  pen  upon  a  small  blackboard  with  a  white, 
thinly-liquid  paint  which  made  marks  that  could  be  easily 
erased,  or  upon  a  white  tablet,  less  than  a  foot  square,  strewn 
with  red  flour,  on  which  they  wrote  the  figures  with  a  small 
stick,  so  that  the  figures  appeared  white  on  a  red  ground."1 
To  be  legible  the  figures  had  to  be  quite  large,  hence  it  became 
necessary  to  devise  schemes  for  the  saving  of  space.  This 
was  accomplished  by  erasing  a  digit  as  soon  as  it  had  done 
its  service.  The  Hindus  were  generally  inclined  to  follow  the 
motion  from  left  to  right,  as  in  writing.  Thus  in  adding  254 
and  663,  they  would  say  2  +  6  =  8,  5  +  6  =  11,  which  changes 
8  to  9,  4  +  3  =  7.  Hence  the  sum  917. 

In  subtraction,  they  had  two  methods  when  "  borrowing " 
became  necessary.  Thus,  in  51  —  28,  they  would  say  8  from 
11  =  3,  2  from  4  =  2;  or  they  would  say  8  from  11  =  3,  3 
from  5  =  2. 

In  multiplication  several  methods  were  in  vogue.  Sometimes 
they  resolved  the  multiplier  into  its  factors,  and  then  multi- 
plied in  succession  by  each  factor.  At  other  times  they  would 
resolve  the  multiplier  into  the  sum  or  the  difference  of  two 
numbers  which  were  easier  multipliers.  In  written  work,  a 
multiplication,  such  as  5  x  57893411,  was  done  thus :  5  x  5=25, 
which  was  written  down  above  the  multiplicand  ;  5  x  7  =  35  ; 
adding  3  to  25  gives  28 ;  erase  the  5  and  in  its  place  write  8. 
We  thus  have  285.  Then,  5x8  =  40;  4  +  5  =  9;  replace  5 
by  9,  and  we  have  2890,  etc.  At  the  close  of  the  operation  the 
work  on  the  tablet  appeared  somewhat  as  follows : 

289467055 
57893411    6 

When  the  multiplier  consisted  of  several  digits,  then  the 
Hindu  operation,  as  described  by  Hankel  (p.  188),  in  case  of 

1  HANKEL,  p.  186. 


HINDUS  97 

324  x  753,  was  as  follows  :   Place  the  left-hand  digit  of  the 
2259        multiplier,  324,  over  units'  place  in  the  multiplicand; 

324  3x7  =  21,  write  this  down ;  3  x  5  =  15 ;  replace  21 
753  by  22 ;  3  x  3  =  9.  At  this  stage  the  appearance  of 
the  work  is  as  here  indicated.  Next,  the  multiplicand  is 
moved  one  place  to  the  right.  2x7  =  14;  in  the  place  where 
the  14  belongs  we  already  have  25,  the  two  together  give  39, 
24096  which  is  written  in  place  of  25 ;  2  x  5  =  10 ;  add  10 

324  to  399,  and  write  409  in  place  of  399 ;  2x3  =  6. 
753  «j<ne  a(jj0ining  ngure  shows  the  work  at  this  stage. 
We  begin  the  third  step  by  again  moving  the  multiplicand  to 
the  right  one  place ;  4  x  7  =  28 ;  add  this  to  09  and  write 
down  37  in  its  place,  etc. 
243972  This  method,  employed  by  Hindus  even  at  the  pres- 

324     ent  time,  economizes  space  remarkably  well,  since  only 

753  a  few  of  all  the  digits  used  appear  on  the  tablet  at  any 
one  moment.  It  was,  therefore,  well  adapted  to  their  small 
tablets  and  coarse  pencils.  The  method  is  a  poor  one,  if  the 
calculation  is  to  be  done  on  paper,  (1)  because  we  cannot 
readily  and  neatly  erase  the  digits,  and  (2)  because,  having 
plenty  of  paper,  it  is  folly  to  save  space  and  thereby  un- 
necessarily complicate  the  process  by  performing  the  addition 
of  each  partial  product,  as  soon  as  formed.  Nevertheless,  we 
find  that  the  early  Arabic  writers,  unable  to  improve  on  the 
Hindu  process,  adopted  it  and  showed  how  it  can  be  carried 
out  on  paper,  viz.,  by  crossing  out  the  digits  (instead  of  eras- 
ing them)  and  placing  the  new  digits  above  the  old  ones.1 

Besides  these,  the  Hindus  had  other  methods,  more  closely 
resembling  the  processes  in  vogue  at  the  present  time.  Thus 
a  tablet  was  divided  into  squares  like  a  chess-board.  Diagonals 
were  drawn.  The  multiplication  of  12  x  735  =  8820  is  ex- 

i  HANKEL,  p.  188. 


98 


A   HISTORY   OF   MATHEMATICS 


hibited  in  the  adjoining  diagram.1     The  manuscripts  extant 
give  no  detailed  information  regarding  the 
Hindu  process  of  division.     It  seems  that 
the  partial   products   were   deducted  by 
erasing  digits  in  the  dividend  and  replac- 
ing them  by  the  new  ones  resulting  from 
the  subtraction.     The  above  scheme  of 
multiplication  was  known  to  the  Arabs  and  may  not  be  of 
Hindu  origin. 

The  Hindus  used  an  ingenious  process  (not  of  Indian 
origin)  for  testing  the  correctness  of  their  computations.  It 
rests  on  the  theorem  that  the  sum  of  the  digits  of  a  number, 
divided  by  9,  gives  the  same  remainder  as  does  the  number 
itself,  divided  by  9.  The  process  of  "  casting  out  the  9's  "  was 
more  serviceable  to  the  Hindus  than  it  is  to  us.  Their  custom 
of  erasing  digits  and  writing  others  in  their  places  made  it 
much  more  difficult  for  them  to  verify  their  results  by  review- 
ing the  operations  performed.  At  the  close  of  a  multiplication 
a  large  number  of  the  digits  arising  during  the  process  were 
erased.  Hence  a  test  which  did  not  call  for  the  examination 
of  the  intermediate  processes  was  of  service  to  them. 

In  the  extant  fragments  of  the  Bakhshdll  arithmetic,  a 
knowledge  of  the  processes  of  computation  is  presupposed. 
In  fractions,  the  numerator  is  written  above  the  denominator 
without  a  dividing  line.  Integers  are  written  as  fractions 
with  the  denominator  1.  In  mixed  expressions  the  integral 

part  is  written  above  the  fraction.     Thus,  1  =  1^.     In  place 

3 
of  our  =  they  used  the  word  pkcdam,  abbreviated  into  pha. 

Addition  was  indicated  by  yu,  abbreviated  from  yuta.     Num- 
bers  to   be    combined   were    often    enclosed    in   a   rectangle. 


Thus,  pha  12 


means    -f-  +  £  =  12.       An    unknown 
1  CANTOR,  I.,  p.  611. 


HINDUS  99 

quantity  is  sunya,  and  is  designated  thus  •  by  a  heavy 
dot.  The  word  sunya  means  "empty,"  and  is  the  word  for 
zero,  which  is  here  likewise  represented  by  a  dot.  This  double 
use  of  the  word  and  dot  rested  upon  the  idea  that  a  position 
is  "empty"  if  not  filled  out.  It  is  also  to  be  considered 
"  empty "  so  long  as  the  number  to  be  placed  there  has  not 
been  ascertained.1 

The  Bakhshali  arithmetic  contains  problems  of  which  some 
are  solved  by  reduction  to  unity  or  by  a  sort  of  false  position. 
EXAMPLE  :  B  gives  twice  as  much  as  A,  C  three  times  as  much 
as  B,  D  four  times  as  much  as  C ;  together  they  give  132 ; 
how  much  did  A  give  ?  Take  1  for  the  unknown  (sunya\ 
then  A  =  1,  B  =  2,  C  =  6,  D  =  24,  their  sum  =  33.  Divide 
132  by  33,  and  the  quotient  4  is  what  A  gave. 

The  method  of  false  position  we  have  encountered  before 
among  the  early  Egyptians.  With  them  it  was  an  instinctive 
procedure ;  with  the  Hindus  it  had  risen  to  a  conscious  method. 
Bhaskara  uses  it,  but  while  the  Bakhshali  document  preferably 
assumes  1  as  the  unknown,  Bhaskara  is  partial  to  3.  Thus, 
if  a  certain  number  is  taken  five-fold,  ^  of  the  product  be 
subtracted,  the  remainder  divided  by  10,  and  ^,  J,  and  \  of  the 
original  number  added,  then  68  is  obtained.  What  is  the  num- 
ber ?  Choose  3,  then  you  get  15, 10, 1,  and  1  +  f  +  f  +  J  =  ^-. 
Then  (68  -H  *£)  3  =  48,  the  answer.2 

A  favourite  method  of  solution  is  that  of  inversion.  With 
laconic  brevity,  Aryabhatta  describes  it  thus:  "Multiplica- 
tion becomes  division,  division  becomes  multiplication;  what 
was  gain  becomes  loss,  what  loss,  gain;  inversion."  Quite 
different  from  this  in  style  is  the  following  problem  of 
Aryabhatta,  which  illustrates  the  method :  "  Beautiful  maiden 
with  beaming  eyes,  tell  me,  as  thou  understands t  the  right 

1  CANTOR,  I.,  pp.  613-615.  2  CANTOR,  I.,  p.  618. 


100  A   HISTORY    OF   MATHEMATICS 

method  of  inversion,  which  is  the  number  which  multiplied  by 
3,  then  increased  by  J  of  the  product,  divided  by  7,  diminished 
by  ^  of  the  quotient,  multiplied  by  itself,  diminished  by  52,  by 
extraction  of  the  square  root,  addition  of  8,  and  division  by  10, 
gives  the  number  2  ?  "  The  process  consists  in  beginning  with 
2  and  working  backwards.  Thus,  (2 . 10  -  8)2  +  52  =  196, 
V196  =  14,  and  14  •  f  •  7  •  f  -:-  3  =  28,  the  answer.1  Here  is 
another  example  taken  from  the  Lilavati:  "The  square  root 
of  half  the  number  of  bees  in  a  swarm  has  flown  out  upon  a 
jessamine-bush,  f  of  the  whole  swarm  has  remained  behind ; 
one  female  bee  flies  about  a  male  that  is  buzzing  within  a  lotus- 
flower  into  which  he  was  allured  in  the  night  by  its  sweet 
odour,  but  is  now  imprisoned  in  it.  Tell  me  the  number  of 
bees."2  Answer  72.  The  pleasing  poetic  garb  in  which  arith- 
metical problems  are  clothed  is  due  to  the  Hindu  practice  of 
writing  all  school-books  in  verse,  and  especially  to  the  fact 
that  these  problems,  propounded  as  puzzles,  were  a  favourite 
social  amusement.  Says  Brahmagupta :  "These  problems  are 
proposed  simply  for  pleasure;  the  wise  man  can  invent  a 
thousand  others,  or  he  can  solve  the  problems  of  others  by  the 
rules  given  here.  As  the  sun  eclipses  the  stars  by  his  brilliancy, 
so  the  man  of  knowledge  will  eclipse  the  fame  of  others  in 
assemblies  of  the  people  if  he  proposes  algebraic  problems,  and 
still  more  if  he  solves  them." 

The  Hindus  were  familiar  with  the  Rule  of  Three,  with  the 
computation  of  interest  (simple  and  compound),  with  alliga- 
tion, with  the  fountain  or  pipe  problems,  and  with  the  sum- 
mation of  arithmetic  and  geometric  series.  Aryabhatta 
applies  the  Eule  of  Three  to  the  problem  —  a  16  year  old 
girl  slave  costs  32  nishkas,  what  costs  one  20  years  old? — and 
says  that  it  is  treated  by  inverse  proportion,  since  "the  value 

1  CANTOR  I.,  p.  618.  a HANKEL,  p.  191. 


of  living  creatures  (slaves  and  cattle)  is  regulated  by  their 
age,"  the  older  being  the  cheaper.1  The  extraction  of  square 
and  cube  roots  was  familiar  to  the  Hindus.  This  was  done 
with  aid  of  the  formulas 

(a  +  6)2  =  a2  +  2  ab  +  62 ;  (a  +  6)3  =  a3  +  3  a26  +  3  ab2  +  63. 

The  Hindus  made  remarkable  contributions  to  Algebra. 
Addition  was  indicated  simply  by  juxtaposition  as  in  Dio- 
phantine  algebra;  subtraction,  by  placing  a  dot  over  the 
subtrahend;  multiplication,  by  putting  after  the  factors,  bha, 
the  abbreviation  of  the  word  bhavita,  "the  product";  division, 
by  placing  the  divisor  beneath  the  dividend ;  square  root,  by 
writing  ka,  from  the  word  karana  (irrational),  before  the 
quantity.  The  unknown  quantity  was  called  by  Brahmagupta 
ydvattdvat.  In  case  of  several  unknown  quantities,  he  gave, 
unlike  Diophantus,  a  distinct  name  and  symbol  to  each.  Un- 
known quantities  after  the  first  were  assigned  the  names  of 
colours,  being  called  the  black,  blue,  yellow,  red,  or  green 
unknown.  The  initial  syllable  of  each  word  was  selected  as 
the  symbol  for  the  unknown.  Thus  yd  meant  a?;  kd  (from 
kdlaka  —  black)  meant  y ;  yd  Jed  bha,  "  x  times  y  " ;  ka  15  ka  10, 
"V15-VI6." 

The  Hindus  were  the  first  to  recognize  the  existence  of 
absolutely  negative  numbers  and  of  irrational  numbers.  The 
difference  between  -f  and  —  numbers  was  brought  out  by 
attaching  to  the  one  the  idea  of  "assets,"  to  the  other  that 
of  "debts,"  or  by  letting  them  indicate  opposite  directions. 
A  great  step  beyond  Diophantus  was  the  recognition  of  two 
answers  for  quadratic  equations.  Thus  Bhaskara  gives  x  =  50 
or  -  5  for  the  roots  of  y?  -  45  x  =  250.  "  But,"  says  he,  "  the 
second  value  is  in  this  case  not  to  be  taken,  for  it  is  inadequate ; 

1  CANTOR!.,  p.  618. 


102 


**3 


,?S*'.'A   HISTOftY   OF  MATHEMATICS 


people  do  not  approve  of  negative  roots."  Thus  negative  roots 
were  seen,  but  not  admitted.  In  the  Hindu  mode  of  solving 
quadratic  equations,  as  found  in  Brahmagupta  and  Aryabhatta, 
it  is  believed  that  Greek  processes  are  discernible.  For 
example,  in  their  works,  as  also  in  Heron  of  Alexandria, 
oo2  -f-  bx  =  c  is  solved  by  a  rule  yielding 


This  rule  is  improved  by  Qridhara,  who  begins  by  multiplying 
the  members  of  the  equation,  not  by  a  as  did  his  predecessors, 
but  by  4  a,  whereby  the  possibility  of  fractions  under  the 
radical  sign  is  excluded.  He  gets 

_  V4  ac  H-  b2  —  b 
2a 

The  most  important  advance  in  the  theory  of  affected  quadratic 
equations  made  in  India  is  the  unifying  under  one  rule  of  the 
three  cases 

bx  =  c,  bx  4-  c  =  aa?2,   ay?  -4-  c  =  bx. 


In  Diophantus  these  cases  seem  to  have  been  dealt  with  sepa- 
rately, because  he  was  less  accustomed  to  deal  with  negative 
quantities.1 

An  advance  far  beyond  the  Greeks  and  even  beyond 
Brahmagupta  is  the  statement  of  Bhaskara  that  "the  square 
of  a  positive,  as  also  of  a  negative  number,  is  positive  ;  that 
the  square  root  of  a  positive  number  is  twofold,  positive  and 
negative.  There  is  no  square  root  of  a  negative  number,  for 
it  is  not  a  square." 

We  have  seen  that  the  Greeks  sharply  discriminated  between 
numbers  and  magnitudes,  that  the  irrational  was  not  recognized 

1  CANTOR,  L,  p.  625. 


ARABS  103 

by  them  as  a  number.  The  discovery  of  the  existence  of 
irrationals  was  one  of  their  profoundest  achievements.  To 
the  Hindus  this  distinction  between  the  rational  and  irrational 
did  not  occur ;  at  any  rate,  it  was  not  heeded.  They  passed 
from  one  to  the  other,  unmindful  of  the  deep  gulf  separating 
the  continuous  from  the  discontinuous.  Irrationals  were  sub- 
jected to  the  same  processes  as  ordinary  numbers  and  were 
indeed  regarded  by  them  as  numbers.  By  so  doing  they 
greatly  aided  the  progress  of  mathematics ;  for  they  accepted 
results  arrived  at  intuitively,  which  by  severely  logical 
processes  would  call  for  much  greater  effort.  Says  Hankel 
(p.  195),  "if  one  understands  by  algebra  the  application  of 
arithmetical  operations  to  complex  magnitudes  of  all  sorts, 
whether  rational  or  irrational  numbers  or  space-magnitudes, 
then  the  learned  Brahmins  of  Hindostan  are  the  real  inventors 
of  algebra." 

In  Bhaskara  we  find  two  remarkable  identities,  one  of  which 
is  given  in  nearly  all  our  school  algebras,  as  showing  how  to 
find  the  square  root  of  a  "binomial  surd."  What  Euclid  in 
Book  X.  embodied  in  abstract  language,  difficult  of  compre- 
hension, is  here  expressed  to  the  eye  in  algebraic  form  and 
applied  to  numbers : 1 


Arabs 

The  Arabs  present  an  extraordinary  spectacle  in  the  history 
of  civilization.      Unknown,  ignorant,  and  disunited  tribes  of 
the  Arabian  peninsula,  untrained  in  government  and  war,  are 
i  HJLNKBL,  p.  194. 


104  A    HISTORY   OF    MATHEMATICS 

in  the  course  of  ten  years  fused  by  the  furnace  blast  of 
religious  enthusiasm  into  a  powerful  nation,  which,  in  one 
century,  extended  its  dominions  from  India  across  northern 
Africa  to  Spain.  A  hundred  years  after  this  grand  march  of 
conquest,  we  see  them  assume  the  leadership  of  intellectual 
pursuits;  the  Moslems  became  the  great  scholars  of  their 
time. 

About  150  years  after  Mohammed's  flight  from  Mecca  to 
Medina,  the  study  of  Hindu  science  was  taken  up  at  Bagdad 
in  the  court  of  Caliph  Almansur.  In  773  A.D.  there  appeared 
at  his  court  a  Hindu  astronomer  with  astronomical  tables  which 
by  royal  order  were  translated  into  Arabic.  These  tables, 
known  by  the  Arabs  as  the  Sindhind,  and  probably  taken  from 
the  Siddhdnta  of  Brahmagupta,  stood  in  great  authority. 
With  these  tables  probably  came  the  Hindu  numerals.  Except 
for  the  travels  of  Albiruni,  we  possess  no  other  evidence  of 
intercourse  between  Hindu  and  Arabic  scholars ;  yet  we  should 
not  be  surprised  if  future  historical  research  should  reveal 
greater  intimacy.  Better  informed  are  we  as  to  the  Arabic 
acquisition  of  Greek  learning.  Elsewhere  we  shall  speak  of 
geometry  and  trigonometry.  Abtfl  Wafd  (940-998)  translated 
the  treatise  on  algebra  by  Diophantus,  one  of  the  last  of 
the  Greek  authors  to  be  brought  out  in  Arabic.  Euclid,  Apol- 
lonius,  and  Ptolemaeus  had  been  acquired  by  the  Arabic  schol- 
ars at  Bagdad  nearly  two  centuries  earlier. 

Of  all  Arabic  arithmetics  known  to  us,  first  in  time  and 
first  in  historical  importance  is  that  of  Muhammed  ibn  Mtisd 
Alchwarizmi  who  lived  during  the  reign  of  Caliph  Al  Mamun 
(813-833).  Like  all  Arabic  mathematicians  whom  we  shall 
name,  he  was  first  of  all  an  astronomer ;  with  the  Arabs  as  with 
the  Hindus,  mathematical  pursuits  were  secondary.  Alchwa- 
rizmt's  arithmetic  was  supposed  to  be  lost,  but  in  1857  a 
Latin  translation  of  it,  made  probably  by  Athelard  of  Bath, 


ARABS  105 

was  found  in  the  library  of  the  University  of  Cambridge.1    The 
arithmetic  begins  with  the  words,  "  Spoken  has  Algoritmi.   Let 
us  give  deserved  praise  to  God,  our  leader  and  defender."   Here 
the  name  of  the  author,  Alchwariznd,  has  passed  into  Algoritmi, 
whence  comes  our  modern  word  algorithm,  signifying  the  art  of 
computing  in  any  particular  way.      Alchwarizmi  is  familiar 
with  the  principle  of  local  value  and  Hindu  processes  of  calcu- 
lation.    According  to  an  Arabic  writer,  his  arithmetic  "  excels 
all  others  in  brevity  and  easiness,  and  exhibits  the  Hindu  intel- 
lect and  sagacity  in  the  grandest  inventions." 2   Both  in  addition 
and  subtraction  Alchwarizmi  proceeds  from  left  to  right,  but  in 
subtraction,  strange  to  say,  he  fails  to  explain  the  case  when 
a  larger  digit  is  to  be  taken  from  a  smaller.    His  multiplication 
is  one  of  the  Hindu  processes,  modified  for  working  on  paper : 
each  partial  product  is  written  over  the  corresponding  digit  in 
the  multiplicand ;  digits  are  not  erased  as  with  the  Hindus,  but 
are  crossed  out.      The  process  of  division  rests  on  the  same 
idea.      The  divisor  is  written  below  the  dividend,  the  quotient 
136    above  it.     The  changes  in  the  dividend  resulting  from 
24      the   subtraction  of  the  partial  products  are  written 
110      above  the  quotient.      For  every  new  step  in  the  divi- 
sion, the  divisor  is  moved  to  the  right  one  place.    The 
143        author  gives  a  lengthy  description  of  the  process  in 
46468    case  of  46468  H-  324  =  143iff ,  which  is  represented  by 
324        Cantor3  by  the  adjoining  model  solution.      This  proc- 
ess of  division  was  used  almost  exclusively  by  early 
European  writers  who  followed  Arabic  models,  and  it 

1  It  was  found  by  Prince  B.  BONCOMPAGNI  under  the  title  "Algoritmi 
de  numero  Indorum."     He  published  it  in  his  book  Trattati  cP  Arit- 
metica,  Rome,  1857. 

2  CANTOR,  L,  713 ;  HANKEL,  p.  256. 

3  CANTOR,  I.,  717.    This  process  of  division  will  be  explained  more  fully 
under  Pacioli. 


106  A    HISTORY    OF    MATHEMATICS 

was  not  extinct  in  Europe  in  the  eighteenth  century.1  Later 
Arabic  authors  modified  Alchwarizmi's  process  and  thereby 
approached  nearer  to  those  now  prevalent.  Alchwarizmi 
explains  in  detail  the  use  of  sexagesimal  fractions. 

Arabic  arithmetics  usually  explained,  besides  the  four  cardi- 
nal operations,  the  process  of  "  casting  out  the  9's "  (called 
sometimes  the  "  Hindu  proof "),  the  rule  of  "  false  position," 
the  rule  of  "  double  position,"  square  and  cube  root,  and  frac- 
tions (written  without  the  fractional  line,  as  by  the  Hindus). 
The  Rule  of  Three  occurs  in  Arabic  works,  sometimes  in  their 
algebras.  It  is  a  remarkable  fact  that  among  the  early  Arabs 
no  trace  whatever  of  the  use  of  the  abacus  can  be  discovered. 
At  the  close  of  the  thirteenth  century,  for  the  first  time,  we 
find  an  Arabic  writer,  Ibn  Albannd,  who  uses  processes  which 
are  a  mixture  of  abacal  and  Hindu  computation.  Ibn  Albanna 
lived  in  Bugia,  an  African  seaport,  and  it  is  plain  that  he  came 
under  European  influences  and  thence  got  a  knowledge  of  the 
abacus.2 

It  is  noticeable  that  in  course  of  time,  both  in  arithmetic 
and  algebra,  the  Arabs  in  the  East  departed  further  and 
further  from  Hindu  teachings  and  came  more  thoroughly  under 
the  influence  of  Greek  science.  This  is  to  be  deplored,  for  on 
these  subjects  the  Hindus  had  new  ideas,  and  by  rejecting 
them  the  Arabs  barred  against  themselves  the  road  of  prog- 
ress. Thus,  Al  Karchl  of  Bagdad,  who  lived  in  the  beginning 
of  the  eleventh  century,  wrote  an  arithmetic  in  which  Hindu 
numerals  are  excluded !  All  numbers  in  the  text  are  written 
out  fully  in  words.  In  other  respects  the  work  is  modelled 
almost  entirely  after  Greek  patterns.  Another  prominent  and 

1  HANKEL,  p.  258. 

2  Whether  the  Arabs  knew  the  use  of  the  abacus  or  not  is  discussed 
also  by  H.  WEISSENBORN,   Einfiihrung  der  jetzigen  Ziffern  in  Europa 
durch  Gerbert,  pp.  5-9. 


ARABS  107 

able  writer,  Abu' I  Wafa,  in  the  second  half  of  the  tenth  cen- 
tury, wrote  an  arithmetic  in  which  Hindu  numerals  find  no 
place.  The  question  why  Hindu  numerals  are  ignored  by 
authors  so  eminent,  is  certainly  a  puzzle.  Cantor  suggests  that 
at  one  time  there  may  have  been  rival  schools,  of  which  one 
followed  almost  exclusively  Greek  mathematics,  the  other 
Indian.1 

The  algebra  of  Alchwarizmi  is  the  first  work  in  which  the 
word  "algebra"  occurs.  The  title  of  the  treatise  is  aldschebr 
walmukdbala.  These  two  words  mean  "  restoration  and  oppo- 
sition." By  "  restoration  "  was  meant  the  transposing  of  neg- 
ative terms  to  the  other  side  of  the  equation;  by  "opposi- 
tion," the  discarding  from  both  sides  of  the  equation  of  like 
terms  so  that,  after  this  operation,  such  terms  appear  only  on 
that  side  of  the  equation  on  which  they  were  in  excess.  Thus, 
5x*  —  2x  =  6  +3x2  passes  by  aldschebr  into  5  a2  =  6  +  2  a; 
+  3  x2 ;  and  this,  by  walmukabala  into  2  x*  =  6  +  2  x.  When 
Alchwarizmi's  aldschebr  walmukdbala  was  translated  into 
Latin,  the  Arabic  title  was  retained,  but  the  second  word  was 
gradually  discarded,  the  firs^t  word  remaining  in  the  form  of 
algebra.  Such  is  the  origin  of  this  word,  as  revealed  by  the 
study  of  manuscripts.  Several  popular  etymologies  of  the 
word,  unsupported  by  manuscript  evidence,  used  to  be  current. 
For  instance,  "  algebra "  was  at  one  time  derived  from  the 
name  of  the  Arabic  scholar,  Dschdbir  ibn  Aflah,  of  Seville, 
who  was  called  Geber  by  the  Latins.  But  Geber  lived  two 
centuries  after  Alchwarizini,  and  therefore  two  centuries  after 
the  first  appearance  of  the  word.2 

Alchwarizmi's  algebra,  like  his  arithmetic,  contains  nothing 

1  CANTOR,  I.,  765. 

2  See  CANTOR,  I.,  719,  721,  722  ;  HANKEL,  p.  248  ;  FELIX  MULLER,  His- 
torisch-etymologische  Studien  uber  Mathematische  Terminologie,  Berlin, 
1887,  pp.  9,  10. 


108  A   HISTORY   OF   MATHEMATICS 

original.  It  explains  the  elementary  operations  and  the  solu- 
tion of  linear  and  quadratic  equations.  Whence  did  the 
author  receive  his  knowledge  of  algebra?  That  he  got  it 
solely  from  Hindu  sources  is  impossible,  for.  the  Hindus  had 
no  such  rules  as  those  of  "restoration"  and  "opposition"; 
they  were  not  in  the  habit  of  making  all  terms  in  an  equa- 
tion positive  as  is  done  by  "restoration."  Alchwarizmi's 
rules  resemble  somewhat  those  of  Diophantus.  But  we  can- 
not conclude  that  our  Arabic  author  drew  entirely  from 
Greek  sources,  for,  unlike  Diophantus,  but  like  the  Hindus,  he 
recognized  two  roots  to  a  quadratic  and  accepted  irrational 
solutions.  It  would  seem,  therefore,  that  the  aldschebr  wal- 
mukdbala  was  neither  purely  Greek  nor  purely  Hindu,  but  was 
a  hybrid  of  the  two,  with  the  Greek  element  predominating. 

In  one  respect  this  and  other  Arabic  algebras  are  inferior  to 
both  the  Hindu  and  the  Diophantine  models :  the  Eastern 
Arabs  use  no  symbols  whatever.  With  respect  to  notation, 
algebras  have  been  divided  into  three  classes  : l  (1)  Rhetorical 
Algebras,  in  which  no  symbols  are  used,  everything  being 
written  out  in  words.  Under  this  head  belong  Arabic  works 
(excepting  those  of  the  later  Western  Arabs),  the  Greek  works 
of  lamblichus  and  Thymaridas,  and  the  works  of  the  early 
Italian  writers  and  of  Kegiomontanus.  The  equation  x*  +  10  x 
=  39  was  indicated  by  Alchwarizmi  as  follows:  "A  square 
and  ten  of  its  roots  are  equal  to  thirty-nine  dirhem ;  that  is,  if 
you  add  to  a  square  ten  roots,  then  this  together  equals  thirty- 
nine." 

(2)  Syncopated  Algebras,  in  which,  as  in  the  first  class,  every- 
thing is  written  out  in  words,  except  that  the  abbreviations  are 
used  for  certain  frequently  recurring  operations  and  ideas. 
Such  are  the  works  of  Diophantus,  those  of  the  later  Western 

l  NESSBLMANN,  pp.  302-306. 


ARABS  109 

Arabs,  and  of  the  later  European  writers  down  to  about  the 
middle  of  the  seventeenth  century  (excepting  Vieta's).  As 
an  illustration  we  take  a  sentence  from  Diophantus,  in  which, 
for  the  sake  of  clearness,  we  shall  use  the  Hindu  numerals 
and,  in  place  of  the  Greek  symbols,  the  corresponding  English 
abbreviations.  If  S.,  N.,  U.,  m.  stand  for  "square,"  "'num- 
ber," "  unity,"  "  minus,"  then  the  solution  of  problem  III.,  7, 
in  Diophantus,  viz.,  to  find  three  numbers  whose  sum  is  a 
square  and  such  that  any  pair  is  a  square,  is  as  follows  :  "  Let 
us  assume  the  sum  of  the  three  numbers  to  be  equal  to  the 
square l  1  S.  2  '  N.  1  U.,  the  first  and  the  second  together  1  S. 
then  the  remainder  2  N.  1  U.  will  be  the  third  number.  Let 
the  second  and  the  third  be  equal  to  1  S.  1  U.  m.  2  N.,  of 
which  the  root  is  IN.  m.  1  U.  Now  all  three  numbers  are 
1  S.  2  N.  1  U. ;  hence  the  first  will  be  4  N.  But  this  and 
the  second  together  were  put  equal  to  1  S.,  hence  the  second 
will  be  1  S.  minus  4  N.  Consequently,  the  first  and  third 
together,  6  N.  1  U.  must  be  a  square.  Let  this  number  be 
121  U.,  then  the  number  becomes  20  U.  Hence  the  first  is 
80  U.,  the  second  320  U.,  the  third  41  U.,  and  they  satisfy 
the  conditions." 

(3)  Symbolic  Algebras,  in  which  all  forms  and  operations 
are  represented  by  a  fully  developed  symbolism,  as,  for  ex- 
ample, x2  -f-  10  x  =  39.  In  this  class  may  be  reckoned  Hindu 
works  as  well  as  European  since  the  middle  of  the  seventeenth 
century. 

From  this  classification  due  to  Nesselmann  the  advanced 
ground  taken  by  the  Hindus  is  brought  to  full  view;  also 
the  step  backward  taken  by  the  early  Arabs.  The  Arabs, 
however,  made  substantial  contributions  to  what  we  may  call 
geometrical  algebra.  Not  only  did  they  (Alchwarizmi,  Al 

1  In  our  notation  the  expressions  given  here  are  respectively, 

,  4x,  x2,  z2-4z,  6x+l. 


110  A   HISTORY   OF   MATHEMATICS 

Karchi)  give  geometrical  proofs,  besides  the  arithmetical,  for 
the  solution  of  quadratic  equations,  but  they  (Al  Mahani,  Abu 
Dscha  'far  Alchazin,  Abu'l  Dschud,  'Omar  Alchaijami)  dis- 
covered a  geometrical  solution  of  cubic  equations,  which  alge- 
braically were  still  considered  insolvable.  The  roots  were 
constructed  by  intersecting  conies.1 

Al  Karchi  was  the  first  Arabic  author  to  give  and  prove 
the  theorems  on  the  summation  of  the  series  : 


I3  +  23  +  33  +  ...  +  n3=  (1  +  2  +  ...  +  ri)2 

Allusion  has  been  made  to  the  fact  that  the  Western  Arabs 
developed  an  algebraic  symbolism.  While  in  the  East  the 
geometric  treatment  of  algebra  had  .  become  the  fashion,  in 
the  West  the  Arabs  elaborated  arithmetic  and  algebra  inde- 
pendently of  geometry.  Of  interest  to  us  is  a  work  by  Alkal- 
sddl  of  Andalusia  or  of  Granada,  who  died  in  1486  or  1477.2 
His  book  was  entitled  Raising  of  the  Veil  of  the  Science  of 
Gubdr.  The  word  "  gubar  "  meant  originally  "  dust,"  and 
stands  here  for  written  arithmetic  with  numerals,  in  contrast 
to  mental  arithmetic.  In  addition,  subtraction,  and  multipli- 
cation, the  result  is  written  above  the  other  figures.  The 
square  root  is  indicated  by  -^,  the  initial  letter  of  the  word 
dschidr,  meaning  "root,"  particularly  "square  root."  Thus, 

V48  =  ^.     The  proportion  7  :  12  =  84  :  x  was  written  ^  .-.  84 

.'.12.  '.7,  the  symbol  for  the  unknown  being  here  probably 
imagined  to  be  the  initial  letter  of  the  word  dschahala,  "not  to 
know."  Observe  that  the  Arabic  writing  is  from  right  to  left. 
In  algebra  proper,  the  unknown  was  expressed  by  the  words 

1  Consult  CANTOR,  I.,  721,  725,  781,  776-778  ;  HANKEL,  pp.  274-280. 

2  CANTOR,  I.,  810-817. 


EUROPE  DURING   THE   MIDDLE   AGES  111 

schai  or  dschidr,  for  which  Alkalsadi  uses  the  abbreviations 
x  =  ^,  v?  =  —o,  <J  for  equality.  Thus,  he  writes  3  ar*  =  12  a; 
-f  63  in  this  way 


This  symbolism  probably  began  to  be  developed  among  the 
Western  Arabs  at  least  as  early  as  the  time  of  Ibn  Albanna 
(born  1252  or  1257).  It  is  of  interest,  when  we  remember 
that  in  the  Latin  translations  made  by  Europeans  this  sym- 
bolism was  imitated. 

Europe  during  the  Middle  Ages 

The  barbaric  nations,  which  from  the  swamps  and  forests  of 
the  North  and  from  the  Ural  Mountains  swept  down  upon 
Europe  and  destroyed  the  Koman  Empire,  were  slower  than 
the  Mohammedans  in  acquiring  the  intellectual  treasures  and 
the  civilization  of  antiquity.  The  first  traces  of  mathematical 
knowledge  in  the  Occident  point  to  a  Koman  origin. 

Introduction  of  Roman  Arithmetic.  —  After  Boethius  and 
Cassiodorius,  mathematical  activity  in  Italy  died  out.  About 
one  century  later,  Isidorus  (570-636),  bishop  of  Seville,  in 
Spain,  wrote  an  encyclopaedia,  entitled  Origines.  It  was 
modelled  after  the.  Roman  encyclopaedias  of  Martinus  Capella 
and  of  Cassiodorius.  Part  of  it  is  taken  up  with  the  quad- 
ruvium.  He  gives  definitions  of  technical  terms  and  their 
etymologies  ;  but  does  not  describe  the  modes  of  computation 
then  in  vogue.  He  divides  numbers  into  odd  and  even,  speaks 
of  perfect  and  excessive  numbers,  etc.,  and  finally  bursts  out 
in  admiration  of  number,  as  follows:  "Take  away  number 
from  all  things,  and  everything  goes  to  destruction."  *  After 
Isidorus  comes  another  century  of  complete  darkness;  then 
i  CANTOR,  I.,  824. 


112  A   HISTOKY    OF   MATHEMATICS 

appears  the  English,  monk,  Bede  the  Venerable  (672-735). 
His  works  contain  treatises  on  the  Computus,  or  the  computa- 
tion of  Easter-time,  and  on  finger-reckoning.  It  appears  that 
a  finger-symbolism  was  then  widely  used  for  calculation.  The 
correct  determination  of  the  time  of  Easter  was  a  problem 
which  in  those  days  greatly  agitated  the  church.  The  desira- 
bility of  having  at  least  one  monk  in  each  monastery  who 
could  determine  the  date  of  religious  festivals  appears  to  have 
Deen  the  greatest  incentive  then  existing  toward  the  study  of 
arithmetic.  "The  computation  of  Easter-time,"  says  Cantor, 
"the  real  central  point  of  time-computation,  is  founded  by 
Bede  as  by  Cassiodorius  and  others,  upon  the  coincidence, 
once  every  nineteen  years,  of  solar  and  lunar  time,  and  makes 
no  immoderate  demands  upon  the  arithmetical  knowledge  of 
the  pupil  who  aims  to  solve  simply  this  problem." l  It  is  not 
surprising  that  Bede  has  but  little  to  say  about  fractions.  In 
one  place  he  mentions  the  Roman  duodecimal  division  into 
ounces. 

The  year  in  which  Bede  died  is  the  year  in  which  the  next 
prominent  thinker,  Alcuin  (735-804)  was  born.  Educated  in 
Ireland,  he  afterwards,  at  the  court  of  Charlemagne,  directed 
the  progress  of  education  in  the  great  Frankish  Empire.  In 
the  schools  founded  by  him  at  the  monasteries  were  taught  the 
psalms,  writing,  singing,  computation  (computus),  and  gram- 
mar. As  the  determination  of  Easter  could  be  of  no  particular 
interest  or  value  to  boys,  the  word  computus  probably  refers 
to  computation  in  general.  We  are  ignorant  of  the  modes  of 
reckoning  then  employed.  It  is  not  probable  that  Alcuin  was 
familiar  with  the  abacus  or  the  apices  of  Boethius.  He  be- 
longed to  that  long  list  of  scholars  of  the  Middle  Ages  and  of 
the  Renaissance  who  dragged  the  theory  of  numbers  into 

1  CANTOR,  L,  831. 


EUROPE   DURING   THE   MIDDLE   AGES  113 

theology.  For  instance,  the  number  of  beings  created  by  God, 
who  created  all  things  well,  is  six,  because  six  is  a  perfect 
number  (being  equal  to  the  sum  of  its  divisors  1,  2,  3) ;  but 
8  is  a  defective  number,  since  its  divisors  1  -f-  2  +  4  <  8,  and, 
for  that  reason,  the  second  origin  of  mankind  emanated  from 
the  number  8,  which  is  the  number  of  souls  said  to  have 
been  in  Noah's  Ark. 

There  is  a  collection  of  "Problems  for  Quickening  the 
Mind"  which  is  certainly  as  old  as  1000  A.D.,  and  possibly 
older.  Cantor  is  of  the  opinion  that  it  was  written  much 
earlier,  and  by  Alcuin.  Among  the  arithmetical  problems  of 
this  collection  are  the  fountain-problems  which  we  have  en- 
countered in  Heron,  in  the  Greek  anthology,  and  among  the 
Hindus.  Problem  No.  26  reads :  A  dog  chasing  a  rabbit, 
which  has  a  start  of  150  feet,  jumps  9  feet  every  time  the  rab- 
bit jumps  7.  To  determine  in  how  many  leaps  the  dog  over- 
takes the  rabbit,  150  is  to  be  divided  by  2.  The  35th  problem 
is  as  follows :  A  dying  man  wills  that  if  his  wife,  being  with 
child,  gives  birth  to  a  son,  the  son  shall  inherit  J  and  the 
widow  \  of  the  property ;  but  if  a  daughter  is  born,  she  shall 
inherit  -£%  and  the  widow  -fy  of  the  property.  How  is  the 
property  to  be  divided  if  both  a  son  and  a  daughter  are  born  ? 
This  problem  is  of  interest  because  by  its  close  resemblance  to 
a  Roman  problem  it  unmistakably  betrays  its  Roman  origin. 
However,  its  solution,  given  in  the  collection,  is  different  from 
the  Roman  solution,  and  is  quite  erroneous.  Some  of  the 
problems  are  geometrical,  others  are  merely  puzzles,  such  as 
the  one  of  the  wolf,  goat,  and  cabbage-head,  which  we  shall 
mention  again.  The  collector  of  these  problems  evidently 
aimed  to  entertain  and  please  his  readers.  It  has  been  re- 
marked that  the  proneness  to  propound  jocular  questions  is 
truly  Anglo-Saxon,  and  that  Alcuin  was  particularly  noted  in 
this  respect.  Of  interest  is  the  title  which  the  collection 


114  A   HISTORY   OF  MATHEMATICS 

bears:  "Problems  for  Quickening  the  Mind."  Do  not  these 
words  bear  testimony  to  the  fact  that  even  in  the  darkness  of 
the  Middle  Ages  the  mind-developing  power  of  mathematics 
was  recognized  ?  Plato's  famous  inscription  over  the  entrance 
of  his  academy  is  frequently  quoted;  here  we  have  the  less 
weighty,  but  significant  testimony  of  a  people  hardly  yet 
awakened  from  intellectual  slumber. 

During  the  wars  and  confusion  which  followed  the  fall  of 
the  empire  of  Charlemagne,  scientific  pursuits  were  abandoned, 
but  they  were  revived  again  in  the  tenth  century,  principally 
through  the  influence  of  one  man,  —  Gerbert.  He  was  born  in 
Aurillac  in  Auvergne,  received  a  monastic  education,  and  en- 
gaged in  study,  chiefly  of  mathematics,  in  Spain.  He  became 
bishop  at  Rheims,  then  at  Ravenna,  and  finally  was  made 
Pope  under  the  name  of  Sylvester  II.  He  died  in  1003,  after 
a  life  involved  in  many  political  and  ecclesiastical  quarrels. 

Gerbert  made  a  careful  study  of  the  writings  of  Boethius,  and 
published  two  arithmetical  works,  —  Rule  of  Computation  on 
the  Abacus,  and  A  Small  Book  on  the  Division  of  Numbers. 
Now  for  the  first  time  do  we  get  some  insight  into  methods  of 
computation.  Gerbert  used  the  abacus,  which  was  probably 
unknown  to  Alcuin.  In  his  younger  days  Gerbert  taught 
school  at  Rheims  —  the  trivium  and  quadruvium  being  the 
subjects,  of  instruction  —  and  one  of  his  pupils  tells  us  that 
Gerbert  ordered  from  his  shield-maker  a  leathern  calculating 
board,  which  was  divided  into  27  columns,  and  that  counters 
of  horn  were  prepared,  upon  which  the  first  nine  numerals 
(apices)  were  marked.  Bemelinus,  a  pupil  of  Gerbert,  de- 
scribes the  abacus  as  consisting  of  a  smooth  board  upon  which 
geometricians  were  accustomed  to  strew  blue  sand,  and  then 
to  draw  their  diagrams.  For  arithmetical  purposes  the  board 
was  divided  into  30  columns,  of  which  three  were  reserved  for 
fractions  while  the  remaining  27  were  divided  into  groups 


EUROPE   DURING    THE   MIDDLE   AGES 


115 


with  three  columns  in  each.  Every  group  of  the  columns  was 
marked  respectively  by  the  letters  C  (centum,  100),  D  (decem, 
10),  and  S  (singularis)  or  M  (monas).  Bernelinus  gives  the 
nine  numerals  used  (the  apices  of  Boethius),  and  then  remarks 
that  the  Greek  letters  may  be  used  in  their  place.1  By  the 
use  of  these  columns  any  number  can  be  written  without  intro- 
ducing the  zero,  and  all  operations  in  arithmetic  can  be  per- 
formed. Indeed,  the  processes  of  addition,  subtraction,  and 
multiplication,  employed  by  the  abacists,  agreed  substantially 
with  those  of  to-day.  The  adjoining  figure  shows  the  multi- 
plication of  4600  by  23.2  The  process  is  as 
follows:  3-6  =  18;  3  x  4  =  12;  2-6  =  12; 
2x4  =  8;  1+2+2  =  5;  remove  the  1,  2, 
2,  and  put  down  5;  1  +  1  +  8  =  10 ;  re- 
move 1, 1,  8,  and  put  down  1  in  the  column 
next  to  the  left.  Hence,  the  sum  105800. 
If  counters  were  used,  then  our  crossing 
out  of  digits  (for  example,  of  the  digits  1, 
2,  2  in  the  fourth  column)  must  be  im- 
agined to  represent  the  removal  of  the  counters  1,  2,  2,  and 
the  putting  of  a  counter  marked  5  in  their  place.  If  the 
numbers  were  written  on  sand,  then  the  numbers  1,  2,  2  were 
erased  and  5  written  instead. 

The  process  of  division  was  entirely  different  from  the 
modern.  So  difficult  has  this  operation  appeared  that  the 
concept  of  a  quotient  may  almost  be  said  to  be  foreign  to 
antiquity.  Gerbert  gave  rules  for  division  which  apparently 
were  framed  to  satisfy  the  following  three  conditions :  (1)  The 
use  of  the  multiplication  table  shall  be  restricted  as  far  as 
possible;  at  least,  it  shall  never  be  required  to  multiply 
mentally  a  number  of  two  digits  by  another  of  one  digit; 


C 

X 

1 

C 

X 

I 

4 

6 

1 

I 

'% 

8 

I 

* 

5 

2 

3 

1  CANTOR,  I.,  881. 


2  FRIEDLEIN,  p.  106. 


116 


A    HISTORY    OF    MATHEMATICS 


(2)  Subtractions  shall  be  avoided  as  much  as  possible  and 
replaced  by  additions ;  (3)  The  operation  shall  proceed  in  a 
purely  mechanical  way,  without  requiring  trials.1  That  it 
should  be  necessary  to  make  such  conditions,  will  perhaps  not 
seem  so  strange,  if  we  recollect  that  monks  of  the  Middle 
Ages  did  not  attend  school  during  childhood  and  learn  the 
multiplication  table  while  the  memory  was  fresh.  Gerbert's 
rules  for  division  are  the  oldest  extant.  They  are  so  brief  as 
to  be  very  obscure  to  the  uninitiated,  but  were  probably 
intended  to  aid  the  memory  by  calling  to  mind  the  successive 
steps  of  the  process.  In  later  manuscripts  they  are  stated 
more  fully.  We  illustrate  this  division  by  the 
example 2  4087  -=-  6  =  681.  The  process  is  a 
kind  of  "complementary  division."  Begin- 
nings of  this  mode  of  procedure  are  found 
among  the  Romans,  but  so  far  as  known  it 
was  never  used  by  the  Hindus  or  Arabs.  It 
is  called  "  complementary "  because,  in  our 
example,  for  instance,  not  6,  but  10—6  or  4 
is  the  number  operated  with.  The  rationale 
of  the  process  may,  perhaps,  be  seen  from  this 
partial  explanation :  4000  -+- 10  =  400,  write 
this  below  as  part  of  the  quotient.  But  10  is 
too  large  a  divisor ;  to  rectify  the  error,  add 
4.400  =  1600.  Then  1000  -  10  =  100,  write 
this  below  as  part  of  the  quotient ;  to  rectify 
this  new  error,  add  4.100  =  400.  Then  600  + 
400  =  1000.  Divide  1000 -r- 10,  and  so  on. 
It  will  be  observed  that  in  complementary 
division,  like  this,  it  was  not  necessary  to 

1  HANKEL,  p.  323. 

2  Quoted  by  FRIEDLEIN,  p.  109.     The  mechanism  of  the  division  is  as 
follows :    Write  down  the   dividend  4087   and  above   it  the   divisor  6. 


I 

c 

X 

I 

4 

6 

I 

p 

\ 

\ 

I 

i 

i 

$ 

i 

$ 

9 

I 

i 

£ 

} 

2 

3 

} 

t 

£ 

$ 

1 

1 

1 

% 

I 

I 

; 

2 

I 

; 

1 

6 

JL 

; 

8 

; 

i 

EUROPE   DURING   THE   MIDDLE   AGES  117 

Know  the  multiplication  table  above  the  5's.1  To  a  modern 
computer  it  would  seem  as  though  the  above  process  of  divi- 
sion were  about  as  complicated  as  human  ingenuity  could 
make  it.  No  wonder  that  it  was  said  of  Gerbert  that  he  gave 
rules  for  division  which  were  hardly  understood  by  the  most 
painstaking  abacists;  no  wonder  that  the  Arabic  method  of 
division,  wlien  first  introduced  into  Europe,  was  called  the 
"golden  division"  (divisio  aurea),  but  the  one  on  the  abacus 
the  "  iron  division  "  (divisio  f errea). 

The  question  has  been  asked,  whence  did  Gerbert  get  his 
abacus  and  his  complementary  division  ?  The  abacus  was 
probably  derived  from  the  works  of  Boethius,  but  the  comple- 
mentary division  is  nowhere  found  in  its  developed  form  before 
the  time  of  Gerbert.  Was  it  mainly  his  invention  ?  From  one 
of  his  letters  it  appears  that  he  had  studied  a  paper  on  multipli- 
cation and  division  by  "  Joseph  Sapiens,"  but  modern  research 
has  as  yet  revealed  nothing  regarding  this  man  or  his  writings.2 

Above  the  6  write  4,  which  is  the  difference  between  10  and  6.  Multiply 
this  difference  4  into  4  in  the  column  I  and  move  the  product  16  to  the 
right  by  one  column  ;  erase  the  4  in  column  I  and  write  it  in  column  C, 
below  the  lower  horizontal  line,  as  part  of  the  quotient.  Multiply  the  1 
in  I  by  4,  write  the  product  4  in  column  C  ;  erase  the  1  and  write  it 
below,  one  column  to  the  right.  Add  the  numbers  in  C,  6  +  4  =  10,  and 
write  1  in  I.  Then  proceed  as  before :  1.4  =  4,  write  it  in  C,  and  write 
1  below.  4.4  =  16  in  C  and  X,  4  below  in  X  ;  1.4  =  4  in  X,  1  below : 
4  +  6  +  8  =  18  in  C  and  X  ;  1.4  =  4  in  X,  1  below  ;  4  +  8  =  12  in  C  and 
X  ;  1.4  =  4  in  X,  1  below  ;  2  +  4  =  6  in  X,  6.4  =  24  in  X  and  I,  6  below  ; 
2.4  =  8  in  I,  2  below ;  8  +  4  +  7  =  19  in  X  and  I ;  1.4  =  4  in  I,  1  below  ; 
9  +  4  =  13  in  X  and  I ;  1.4  =  4  in  I,  1  below  ;  3  +  4  =  7.  Dividing  7  by 
6  goes  1  and  leaves  1.  Write  the  1  in  I  above  and  also  below.  Add  the 
digits  in  the  columns  below  and  the  sum  681  is  the  answer  sought,  i.e. 
4087  -4-  6  =  681,  leaving  the  remainder  1. 

1  For  additional  examples  of  complementary  division  see  FRIEDLEIN, 
pp.  109-124 ;  GiJNTHER,  Math.  Unterr.  im  d.  Mittela.,  pp.  102-106. 

2  Consult    H.    WEISSENBORN,    Einfuhrung    tier  jetzigen    Ziffern    in 
Europa,  Berlin,  1892. 


118  A   HISTORY  OF  MATHEMATICS 

In  course  of  the  next  five  centuries  the  instruments  for 
abacal  computation  were  considerably  modified.  Not  only  did 
the  computing  tables,  strewn  with  sand,  disappear,  but  also 
Gerbert's  abacus  with  vertical  columns  and  marked  counters 
(apices).  In  their  place  there  was  used  a  calculating  board 
with  lines  drawn  horizontally  (from  left  to  right)  and  with 
counters  all  alike  and  unmarked.  Its  use  is  explained  in  the 
first  printed  arithmetics,  and  will  be  described  under  Recorde. 
The  new  instrument  was  employed  in  Germany,  France, 
England,  but  not  in  Italy.1 

Translation  of  Arabic  Manuscripts.  —  Among  the  transla- 
tions which  were  made  in  the  period  beginning  with  the 
twelfth  century  is  the  arithmetic  of  Alchwarizmi  (probably 
translated  by  Athelard  of  Bath),  the  algebra  of  Alchwarizmi 
(by  Gerard  of  Cremona  in  Lombardy)  and  the  astronomy  of  Al 
Battani  (by  Plato  of  Tivoli).  John  of  Seville  wrote  a  liber 
alghoarismi,  compiled  by  him  from  Arabic  authors.  Thus  Ara- 
bic arithmetic  and  algebra  acquired  a  foothold  in  Europe. 
Arabic  or  rather  Hindu  methods  of  computation,  with  the. 
zero  and  the  principle  of  local  value,  began  to  displace  the 
abacal  modes  of  computation.  But  the  victory  of  the  new 
over  the  old  was  not  immediate.  The  struggle  between  the 
two  schools  of  arithmeticians,  the  old  abacistic  school  and  the 
new  algoristic  school,  was  incredibly  long.  The  works  issued 
by  the  two  schools  possess  most  striking  differences,  from 
which  it  would  seem  clear  that  the  two  parties  drew  from  in- 
dependent sources,  and  yet  it  is  argued  by  some  that  Gerbert 
got  his  apices  and  his  arithmetical  knowledge,  not  from 
Boethius,  but  from  the  Arabs  in  Spain,  and  that  part  or  the 
whole  of  the  geometry  of  Boethius  is  a  forgery,  dating  from 
the  time  of  Gerbert.  If  this  were  the  case,  then  we  should 

i  See  CANTOR,  Vol.  II.,  2nd  Edition,  1900,  pp.  216,  217  ;  see  also  GER- 
UARDT,  Geschichte  der  Mathematik  in  Deutschland,  Miinchen,  1877,  p.  29. 


EUROPE   DURING    THE   MIDDLE   AGES  119 

expect  the  writings  of  Gerbert  to  betray  Arabic  sources,  as  do 
those  of  John  of  Seville.  But  no  points  of  resemblance  are 
found.  Gerbert  could  not  have  learned  from  the  Arabs  the 
use  of  the  a,bacus,  because  we  possess  no  reliable  evidence  that 
the  Arabs  ever  used  it.  The  contrast  between  algorists  and 
abacists  consists  in  this,  that  unlike  the  latter,  the  former 
mention  the  Hindus,  use  the  term  algorism,  calculate  with  the 
zero,  and  do  not  employ  the  abacus.  The  former  teach  the 
extraction  of  roots,  the  abacists  do  not;  the  algorists  teach 
sexagesimal  fractions  used  by  the  Arabs,  while  the  abacists 
employ  the  duodecimals  of  the  Romans. 

The  First  Awakening.  —  Towards  the  close  of  the  twelfth 
century  there  arose  in  Italy  a  man  of  genuine  mathematical 
power.  He  was  not  a  monk,  like  Bede,  Alcuin,  and  Gerbert, 
but  a  business  man,  whose  leisure  hours  were  given  to  mathe- 
matical study.  To  Leonardo  of  Pisa,  also  called  Fibonacci,  or 
Fibonaci,  we  owe  the  first  renaissance  of  mathematics  on 
Christian  soil.  When  a  boy,  Leonardo  was  taught  the  use 
of  the  abacus.  In  later  years,  during  his  extensive  travels 
in  Egypt,  Syria,  Greece,  and  Sicily,  he  became  familiar  with 
various  modes  of  computation.  Of  the  several  processes 
he  found  the  Hindu  unquestionably  the  best.  After  his 
return  home,  he  published  in  1202  a  Latin  work,  the  liber 
abaci.  A  second  edition  appeared  in  1828.  While  this  book 
contains  pretty  much  the  entire  arithmetical  and  algebraical 
knowledge  of  the  Arabs,  it  demonstrates  its  author  to  be  more 
than  a  mere  compiler  or  slavish  imitator.  The  liber  abaci 
begins  thus  :  "The  nine  figures  of  the  Hindus  are  9,  8,  7,  6,  5, 
4,  3,  2,  1.  With  these  nine  figures  and  with  this  sign,  0, 
which  in  Arabic  is  called  sifr,  any  number  may  be  written." 
The  Arabic  sifr  (sifra  =  empty)  passed  into  the  Latin  zephirum 
and  the  English  cipher.  If  it  be  remembered  that  the  Afabs 
wrote  from  right  to  left,  it  becomes  evident  how  Leonardo, 


120  A   HISTORY   OF   MATHEMATICS 

in  the  above  quotation,  came  to  write  the  digits  in  descend- 
ing rather  than  ascending  order;  it  is  plain  also  how  he 
came  to  write  ^182  instead  of  1821.  The  liber  abaci  is  the 
earliest  work  known  to  contain  a  recurring  series.1  Interest- 
ing is  the  following  problem  of  the  seven  old  women,  because 
it  is  given  (in  somewhat  different  form)  by  Ahmes,  3000  years 
earlier  :  Seven  old  women  go  to  Rome,  each  woman  has  seven 
mules,  each  mule  carries  seven  sacks,  each  sack  contains  seven 
loaves,  with  each  loaf  are  seven  knives,  each  knife  rests  in 
seven-  sheaths.  What  is  the  sum  total  of  all  named  ?  Am. 
137256.2  Leonardo's  treatise  was  for  centuries  the  storehouse 
from  which  authors  drew  material  for  their  arithmetical  and 
algebraical  books.  Leonardo's  algebra  was  purely  "  rhetor- 
ical " ;  that  is,  devoid  of  all  algebraic  symbolism. 

Leonardo's  fame  spread  over  Italy,  and  Emperor  Frederick 
II.  of  Hohenstaufen  desired  to  meet  him.  The  presentation 
of  the  celebrated  algebraist  to  the  great  patron  of  learning  was 
accompanied  by  a  famous  scientific  tournament.  John  of 
Palermo,  an  imperial  notary,  proposed  several  problems  which 
Leonardo  solved  promptly.  The  first  was  to  find  the  number  a?, 
such  that  or2 +  5  and  a?  —  5  are  each  square  numbers.  The 
answer  is  x  =  3& ;  for  (3 A)8  +  5  =  (4^)" ;  (3T%)2  -  5  =  (2^) 2. 
The  Arabs  had  already  solved  similar  problems,  but  some  parts 
of  Leonardo's  solution  seem  original  with  him.  The  second 
problem  was  the  solution  of  or*  +  2  y?  -f  10  x  =  20.  The  general 
algebraic  solution  of  cubic  equations  was  unknown  at  that 
time,  but  Leonardo  succeeded  in  approximating  to  one  of  the 
roots.  He  gave  x  =  I°22'7"4"'33iv4v40vi,  the  answer  being 
thus  expressed  in  sexagesimal  fractions.  Converted  into  deci- 
mals, this  value  furnishes  figures  correct  to  nine  places.  These 
and  other  problems  solved  by  Leonardo  disclose  brilliant 

1  CANTOR,  II.,  26.  2  CANTOR,  II.,  26. 


EUROPE   DURING   THE   MIDDLE   AGES  121 

talents.  His  geometrical  writings  will  be  touched  upon 
later. 

In  Italy  the  Hindu  numerals  were  readily  accepted  by  the 
enlightened  masses,  but  at  first  rejected  by  the  learned  circles. 
Italian  merchants  used  them  as  early  as  the  thirteenth  cen- 
tury ;  in  1299  the  Florentine  merchants  were  forbidden  the 
use  of  the  Hindu  numerals  in  bookkeeping,  and  ordered  either 
to  use  the  Eoman  numerals  or  to  write  numbers  out  in  words.1 
The  reason  for  this  decree  lies  probably  in  the  fact  that  the 
Hindu  numerals  as  then  employed  had  not  yet  assumed  fixed, 
definite  shapes,  and  the  variety  of  forms  for  certain  digits 
sometimes  gave  rise  to  ambiguity,  misunderstanding,  and  fraud. 
In  our  own  time,  even,  sums  of  money  are  always  written  out 
in  words  in  case  of  checks  or  notes.  Among  the  Italians  are 
evidences  of  an  early  maturity  of  arithmetic.  Says  Peacock, 
"The  Tuscans  generally,  and  the  Florentines  in  particular, 
whose  city  was  the  cradle  of  the  literature  and  arts  of  the  thir- 
teenth and  fourteenth  centuries,  were  celebrated  for  their  know- 
ledge of  arithmetic ;  the  method  of  bookkeeping,  which  is  called 
especially  Italian,  was  invented  by  them;  and  the  operations 
of  arithmetic,  which  were  so  necessary  to  the  proper  conduct 
of  their  extensive  commerce,  appear  to  have  been  cultivated 
and  improved  by  them  with  particular  care;  to  them  we  are 
indebted  ...  for  the  formal  introduction  into  books  of  arith- 
metic, under  distinct  heads,  of  questions  in  the  single  and 
double  rule  of  three,  loss  and  gain,  fellowship,  exchange,  sim- 
ple interest,  discount,  compound  interest,  and  so  on."2 

In  Germany,  France,  and  England,  the  Hindu  numerals 
were  scarcely  used,  until  after  the  middle  of  the  fifteenth 
century.3  A  small  book  on  Hindu  arithmetic,  entitled  De  arte 
numerandi,  called  also  Algorismus,  was  read,  mainly  in  France 

1  HANKEL,  p.  341.  2  PEACOCK,  p.  414.  3  See  UNGEB,  p.  14. 


122  A   HISTORY   OF   MATHEMATICS 

and  Italy,  for  several  centuries.  It  is  usually  ascribed  to  John 
Halifax,  also  called  Sacrobosco,  or  Holywood,  who  was  born 
in  Yorkshire,  and  educated  at  Oxford,  but  who  afterwards  set- 
tled in  Paris  and  taught  there  until  his  death,  in  1256.  The 
booklet  contains  rules  without  proofs  and  without  numerical 
examples ;  it  ignores  fractions.  It  was  printed  in  1488  and 
many  times  later.  According  to  De  Morgan  it  is  the  "  first 
arithmetical  work  ever  printed  in  a  French  town  (Strasbourg)."  * 
Here  and  there  some  of  our  modern  notions  were  anticipated 
by  writers  of  the  Middle  Ages.  For  example,  Nicole  Oresme,  a 
bishop  in  Normandy  (about  1323-1382),  first  conceived  the 
notion  of  fractional  powers,  afterwards  rediscovered  by  Stevin, 
and  suggested  a  notation  for  them.  Thus,2  since  43  =  64  and 
V64  =  8,  it  follows  that  41*  =  8.  In  Oresme's  notation  41*  is 


expressed,    1  p  •  -  4,  or  ^—  -  4.     Such  suggestions  to  the  con- 


trary notwithstanding,  the  fact  remains  that  the  fourteenth  and 
fifteenth  centuries  brought  forth  comparatively  little  in  the 
way  of  original  mathematical  research.  There  were  numerous 
writers,  but  their  scientific  efforts  were  vitiated  by  the  methods 
of  scholastic  thinking. 

GEOMETRY  AND  TRIGONOMETRY 
Hindus 

Our  account  of  Hindu  geometrical  research  will  be  very 
brief ;  for,  in  the  first  place,  like  the  Egyptians  and  Romans, 
the  Hindus  never  possessed  a  science  of  geometry;  in  the 
second  place,  unlike  the  Egyptians  and  Romans,  they  do  not 

iSee  Biblioth.  Mathem.,  1894,  pp.  73-78;  also  1895,  pp.  36-37; 
CANTOR,  II.,  pp.  88-91;  "  De  arte  numerandi"  was  reprinted  last  by 
J.  0.  HALLIWELL,  in  Kara  Mathematics  1839. 

2  CANTOR,  II.,  133. 


HINDUS 


123 


figure  in  geometrical  matters  as  the  teachers  of  other  nations. 
Their  early  geometry  was  original,  their  later  geometry 
came  from  Greece  and  China.  Brahmagupta  gives  the 
"Heronic  Formula"  for  the  area  of  a  triangle.  He  also 
gives  the  proposition  of  Ptolemseus,  that  the  product  of  the 
diagonals  of  a  quadrilateral  is  equal  to  the  sum  of  the  product 
of  the  opposite  sides,  and  adds  new  theorems  of  his  own  on 
quadrilaterals  inscribed  in  a  circle.  The  calculation  of  areas 
forms  the  chief  part  of  Hindu  geometry.  Aryabhatta  gives 
7r  =  fUU-  Interesting  is 
Bhaskara's  proof  of  the 
theorem  of  the  right  tri- 
angle. He  draws  a  right 
triangle  four  times  in  the 
square  of  the  hypotenuse,  so  that  in  the  middle  there  remains 
a  square  whose  side  equals  the  difference  between  the  two 
sides  of  the  right  triangle.  Arranging  this  small  square  and 
the  four  triangles  in  a  different  way,  they  are  seen,  together, 
to  make  up  the  sum  of  the  squares  of  the  two  sides.  "Be- 
hold," says  Bhaskara,  without  adding  another  word  of  ex- 
planation. Eigid  forms  of  demonstration  are  unusual  with 
the  Hindus.  This  proof  is  given  much  earlier  by  a  Chinese 
writer.  Pythagoras7  proof  may  have  been  like  this.  In 
another  place  Bhaskara  gives  a  second  demonstration  of  this 
theorem  by  drawing  from  the  vertex  of  the  right  angle  a 
perpendicular  to  the  hypotenuse,  and  then  suitably  manipulat- 
ing the  proportions  yielded  by  the  similar  triangles.  This  proof 
was  unknown  in  Europe  until  it  was  rediscovered  by  Wallis. 

More  successful  were  the  Hindus  in  the  cultivation  of 
trigonometry.  As  with  the  Greeks,  so  with  them,  it  was 
valued  merely  as  a  tool  in  astronomical  research.  Like  the 
Babylonians  and  Greeks,  they  divide  the  circle  into  360  de- 
grees and  21,600  minutes.  Taking  7r=3.1416,  and  2  7rr=21,600, 


124 


A   HISTOKY   OF   MATHEMATICS 


they  got  r  =  3438 ;  that  is,  the  radius  contained  nearly  3438 
of  these  circular  parts.  This  step  was  not  Grecian.  Some 
Greek  mathematicians  would  have  had  scruples  about  measur- 
ing a  straight  line  by  a  part  of  a  curve.  With  Ptolemy  the 
division  of  the  radius  into  sexagesimal  parts  was  independent 
of  the  division  of  the  circumference ;  no  common  unit  of 
measure  was  selected.  The  Hindus  divided  each  quadrant 
into  24  equal  parts,  so  that  each  of  these  parts  contained  225 
out  of  the  21,600  units.  A  vital  feature  of  Hindu  trigonometry 
is  that  they  did  not,  like  the  Greeks,  reckon  with  the  whole 
chord  of  double  a  given  arc,  but  with  the  sine  of  the  arc  (i.e. 
half  the  chord  of  double  the  arc)  and  with  the  versed  sine  of 
the  arc.  The  entire  chord  AB  was  called 
by  the  Brahmins  jyd  or  jiva,  words  which 
meant  also  the  cord  of  a  hunter's  bow. 
For  AC}  or  half  the  chord,  they  used  the 
words  jydrdha  or  ardhajyd,  but  the  names 
of  the  whole  chord  were  also  used  for 
brevity.  It  is  interesting  to  trace  the  his- 
tory of  these  words.  The  Arabs  transliterated  jivd  or  jiva  into 
dschlba.  For  this  they  afterwards  used  the  word  dschaib,  of 
nearly  the  same  form,  meaning  "  bosom."  This,  in  turn,  was 
translated  into  Latin,  as  sinus,  by  Plato  of  Tivoli.  Thus  arose 
the  word  sine  in  trigonometry.  For  "versed  sine,"  the  Hindus 
used  the  term  utkramajyd,  for  "cosine,"  kotijyd.1  Observe 
that  the  Hindus  used  three  of  our  trigonometric  functions, 
while  the  Greeks  considered  only  the  chord. 

The  Hindus  computed  a  table  of  sines  by  a  theoretically 
simple  method.  The  sine  of  90°  was  equal  to  the  radius,  or 
3438;  the  chord  of  an  arc  AB  of  60°  was  also  3438,  therefore 
half  this  chord  AC,  or  the  sine  of  30°,  was  1719.  Applying  the 


B 


CANTOR,  I.,  658,  736. 


ARABS  125 


formula  sin2a--f-cos2a=r2,  and  observing  that  sin  45°=  cos  45°, 

& 
they  obtained  sin  45°  =•%/—  =  2431.     Substituting  for  cos  a  its 

\  2 

equal  sin  (90  —  a),  and  making  a  =  60°,  they  obtained 


With  the  sines  of  90,  60,  45  as  starting-points,  they  reckoned 
the  sines  of  half  the  angles  by  the  formula  versin  2  a  =2  sin2  a, 
thus  obtaining  the  sines  of  22°  30',  15°,  11°  15',  7°  30',  3°4£'. 
They  now  figured  out  the  sines  of  the  complements  of  these 
angles,  namely,  the  sines  of  86°  15',  82°  30',  78°  45',  75°,  67°  30'; 
then  they  calculated  the  sines  of  half  these  angles;  thereof 
their  complements,  and  so  on.  By  this  very  simple  process 
they  got  the  sines  of  all  the  angles  at  intervals  of  3°45'.1 
No  Indian  treatise  on  the  trigonometry  of  the  triangle  is 
extant.  In  astronomical  works,  there  are  given  solutions  of 
plane  and  spherical  right  triangles.  Scalene  triangles  were 
divided  up  into  right  triangles,  whereby  all  ordinary  compu- 
tations could  be  carried  out.  As  the  table  of  sines  gave  the 
values  for  angles  at  intervals  of  3|  degrees,  the  sines  of  inter- 
vening angles  had  to  be  found  by  interpolation.  Astronomical 
observations  and  computations  possessed  only  a  passable 
degree  of  accuracy.2 

Arabs 

The  Arabs  added  hardly  anything  to  the  ancient  stock  of 
geometrical  knowledge.  Yet  they  play  an  all-important  role 
in  mathematical  history;  they  were  the  custodians  of  Greek 
and  Oriental  science,  which,  in  due  time,  they  transmitted  to 

1  A.  ARNETH,  Die  Geschichte  der  reinen  Mathematik,  Stuttgart,  1852, 
pp.  172,  173.     This  work  we  shall  cite  as  ARNETH.     See,  also,  HANKEL, 
p.  217. 

2  ARNETH,  p.  174. 


126  A   HISTORY    OF   MATHEMATICS 

the  Occident.  The  starting-point  for  all  geometric  study 
among  the  Arabs  was  the  Elements  of  Euclid.  Over  and  over 
again  was  this  great  work  translated  by  them  into  the  Arabic 
tongue.  Imagine  the  difficulties  encountered  in  such  a  trans- 
lation. Here  we  see  a  people,  just  emerged  from  barbarism, 
untrained  in  mathematical  thinking,  and  with  limited  facilities 
for  the  accurate  study  of  languages.  Where  was  to  be  found 
the  man,  who,  without  the  aid  of  grammars  and  dictionaries, 
had  become  versed  in  both  Greek  and  Arabic,  and  was  at 
the  same  time  a  mathematician  ?  How  could  highly  refined 
scientific  thought  be  conveyed  to  undeveloped  minds  by  an 
undeveloped  language  ?  Certainly  it  is  not  strange  that  sev- 
eral successive  efforts  at  translation  had  to  be  made,  each 
translator  resting  upon  the  shoulders  of  his  predecessor. 

Arabic  rulers  wisely  enlisted  the  aid  of  Greek  scholars.  In 
Syria  the  sciences,  especially  philosophy  and  medicine,  were 
cultivated  by  Greek  Christians.  Celebrated  were  the  schools 
at  Antioch  and  Emesa,  and  the  Nestorian  school  at  Edessa. 
After  the  sack  and  ruin  of  Alexandria,  in  640,  they  became  the 
chief  repositories  in  the  East  of  Greek  learning.  Euclid's 
Elements  were  translated  into  Syriac.  From  Syria  Greek 
Christians  were  called  to  Bagdad,  the  Mohammedan  capital. 
During  the  time  of  the  Caliph  Hdrtin  ar-Rasclild  (786-809)  was 
made  the  first  Arabic  translation  of  Ptolemy's  Almagest;  also 
of  Euclid's  Elements  (first  six  books)  by  Haddschddsch  ibn 
Jinsuf  ibn  Matar.1  He  made  a  second  translation  under  the 
Caliph  Al  Mamun  (813-833).  This  caliph  secured  as  a  con- 
dition, in  a  treaty  of  peace  with  the  emperor  in  Constanti- 

1  CANTOR,  I.,  702  ;  Biblioth.  Mathem.,  1892,  p.  65.  An  account  of  trans- 
lators and  commentators  on  Euclid  was  given  by  Ibn  Abi  Ja'kub  an-Nadim 
in  his  Fihrist,  an  important  bibliographical  work  published  in  Arabic  in 
987.  See  a  German  translation  by  H.  SUTER,  in  the  Zeitschr.  fur  Math.  u. 
Phys.,  1892,  Supplement,  pp.  3-87. 


ARABS  127 

nople,  a  large  number  of  Greek  manuscripts,  which  he  ordered 
translated  into  Arabic.  Euclid's  Elements  and  the  Sphere 
and  Cylinder  of  Archimedes  were  translated  by  Ab&  Ja'Mb 
IshdJc  ibn  Hunain,  under  the  supervision  of  his  father  Hunain 
ibn  Ishdk.1  These  renderings  were  unsatisfactory ;  the  trans- 
lators, though  good  philologians,  were  poor  mathematicians. 
At  this  time  there  were  added  to  the  thirteen  books  of  the 
Elements  the  fourteenth,  by  Hypsicles  (?),  and  the  fifteenth  by 
Damascius  (?).  It  remained  for  Tdbit  ibn  Kurrali  (836-901)  to 
bring  forth  an  Arabic  Euclid  satisfying  every  need.  Among 
other  important  translations  into  Arabic  were  the  mathemati- 
cal works  of  Apollonius,  Archimedes,  Heron,  and  Diophantus. 
Thus,  in  course  of  one  century,  the  Arabs  gained  access  to  the 
vast  treasures  of  Greek  science. 

A  later  and  important  Arabic  edition  of  Euclid's  Elements 
was  that  of  the  gifted  Naslr  Eddln  (1201-1274),  a  Persian 
astronomer  who  persuaded  his  patron  Hulagu  to  build  him  and 
his  associates  a  large  observatory  at  Maraga.  He  tried  his 
skill  at  a  proof  of  the  parallel-postulate.  In  all  such  attempts, 
some  new  assumption  is  made  which  is  equivalent  to  the  thing 
to  be  proved.  Thus  Nasir  Eddin  assumes  that  if  AB  is  J_  to 
CD  at  (7,  and  if  another  straight  line  EDF  makes  the  angle 
EDO  acute,  then  the  perpendiculars  to  AB,  comprehended 
between  AB  and  EF,  and  drawn  on  the  side  of  CD  toward  E, 
are  shorter  and  shorter,  the  further  they  are  from  CD.2  It  is 
difficult  to  see  how  in  any  case  this  can  be  otherwise,  unless 
one  looks  with  the  eyes  of  Lobatchewsky  or  Bolyai.  Nasir 
Eddin's  "proof"  had  some  influence  on  the  later  development 
of  the  theory  of  parallels.  His  edition  of  Euclid  was  printed 
in  Arabic  at  Rome  in  1594  and  his  "  proof "  was  brought  out 

1  CANTOR,  I.,  703. 

2  The  proof  is  given  by  KASTNER,  I.,  375-381  and  in  part,  in  Biblioth. 
Mathem.,  1892,  p.  5. 


128  A   HISTORY   OF   MATHEMATICS 

in  Latin  translation  by  Wallis  in  1651. l  Of  interest  is  a  new 
proof  of  the  Pythagorean  Theorem  which  Nasir  Eddin  adds  to 
the  Euclidean  proof.2  An  earlier  demonstration,  for  the  special 
case  of  an  isosceles  right  triangle,3  is  given  by  Muhammed  ibn 
Milsd  Alchwarizml  who  lived  during  the  reign  of  Caliph  Al 
Mamun,  in  the  early  part  of  the  ninth  century.  Alchwarizmi's 
meagre  treatment  of  geometry,  as  contained  in  his  work  on 
Algebra,  is  the  earliest  Arabic  effort  in  this  science.  It 
bears  unmistakable  evidence  of  Hindu  influences.  Besides  the 
value  7T  =  3^,  it  contains  also  the  Hindu  values  TT  =  VlO  and 
7r  =  |m|.  In  later  Arabic  works,  Hindu  geometry  hardly 
ever  shows  itself ;  Greek  geometry  held  undisputed  sway.  In 
a  book  by  the  sons  of  Mtisd  ibn  Sclidkir  (who  in  his  youth  was 
a  robber)  is  given  the  Heronic  Formula  for  the  area  of  a  tri- 
angle. A  neat  piece  of  research  is  displayed  in  the  "  geometric 
constructions  "  by  Abtfl  Wafd  (940-998),  a  native  of  Buzshan 
in  Chorassan.  He  improved  the  theory  of  draughting  by  show- 
ing how  to  construct  the  corners  of  the  regular  polyedrons  on 
the  circumscribed  sphere.  Here,  for  the  first  time,  appears 
the  condition  which  afterwards  became  very  famous  in  the 
Occident,  that  the  construction  be  effected  with  a  single  open- 
ing of  the  compasses. 

The  best  original  work  done  by  the  Arabs  in  mathematics 

1  WALLIS,  Opera,  II.,  669-673. 

2  See  H.  SOTER,  in  Biblioth.  Mathem.,  1892,  pp.  3  and  4.   In  Hoffmann's 
and  Wipper's  collections  of  proofs  for  this  theorem,  Nasir  Eddin's  proof 
is  given  without  any  reference  to  him. 

3  Some  Arabic  writers,  Beha  Eddin  for  instance,  called  the  Pythagorean 
Theorem  the  "figure  of  the  bride."     This  romantic  appellation  originated 
probably  in  a  mistranslation  of  the  Greek  word  v-bp.^-,  applied  to  the 
theorem  by  a  Byzantine  writer  of  the  thirteenth  century.     This  Greek 
word  admits  of  two  meanings,  "  bride"  and  "  winged  insect."     The  fig- 
ure of  a  right  triangle  with  its  three  squares  suggests  an  insect,  but  Beha 
Eddin  apparently  translated  the  word  as  "  bride."     See  PAUL  TANNERY, 
in  I? Intermediate  des  Mathematicieus,  1894,  T.  I.,  p.  254. 


ARABS  129 

is  the  geometric  solution  of  cubic  equations  and  the  develop- 
ment of  trigonometry.  As  early  as  773  Caliph  Almansur  came 
into  possession  of  the  Hindu  table  of  sines,  probably  taken 
from  Brahmagupta's  Siddhdnta.  The  Arabs  called  the  table 
the  Sindhind  and  held  it  in  high  authority.  They  also  came 
into  early  possession  of  Ptolemy's  Almagest  and  of  other  Greek 
astronomical  works.  Mtihammed  ibn  M&sd  Alcliwarizml  was 
engaged  by  Caliph  Al  Mamun  in  making  extracts  from  the 
Sindhind,  in  revising  the  tables  of  Ptolemy,  in  taking  obser- 
vations at  Bagdad  and  Damascus,  and  in  measuring  the  degree 
of  the  earth's  meridian.  Remarkable  is  the 
derivation,  by  Arabic  authors,  of  formulae 
in  spherical  trigonometry,  not  from  the 
"rule  of  six  quantities  of  Menelaus,"  as 
previously,  but  from  the  "rule  of  four 
quantities."  This  is:  If  PPl  and  QQl  be 
two  arcs  of  great  circles  intersecting  in  A,  and  if  PQ  and 
P!^!  be  arcs  of  great  circles  drawn  perpendicular  to  QQ^  then 
we  have  the  proportion 

sin  AP :  sin  PQ  =  sin  AP^ :  sin  PiQi- 

This  departure  from  the  time-honoured  procedure  adopted  by 
Ptolemy  was  formerly  attributed  to  Dschdbir  ibn  Ajlah,  but 
recent  study  of  Arabic  manuscripts  indicates  that  the  transi- 
tion from  the  "rule  of  six  quantities"  to  the  "rule  of  four 
quantities  "  was  possibly  effected  already  by  Tdbit  ibn  Kurrah l 
(836-901),  the  change  being  adopted  by  other  writers  who 
preceded  Dschabir  ibn  Aflah.2 

Foremost  among  the  astronomers  of  the  ninth  century  ranked 

1  H.  SDTER,  in  Biblioth.  Mathem.,  1893,  p.  7. 

2  For  the  mode  of  deriving  formulae  for  spherical  right  triangles,  ac- 
cording to  Ptolemy,  also  according  to  Dschabir  ibn  Aflah  and  his  Arabic 
predecessors,  see  HANKEL,  pp.  285-287  ;  CANTOR,  I.,  794. 

K 


130  A   HISTORY   OF   MATHEMATICS 

Al  Battdni,  called  Albategnius  by  the  Latins.  Battan  in  Syria 
was  his  birth-place.  His  work,  De  scientia  stellarum,  was 
translated  into  Latin  by  Plato  Tiburtinus,  in  the  twelfth 
century.  In  this  translation  the  Arabic  word  dschiba,  from 
the  Sanskrit  fiva,  is  said  to  have  been  rendered  by  the  word 
sinus;  hence  the  origin  of  "sine."  Though  a  diligent  student 
of  Ptolemy,  Al  Battani  did  not  follow  him  altogether.  He 
took  an  important  step  for  the  better,  when  he  introduced  the 
Indian  "  sine  "  or  half  the  chord,  in  place  of  the  ivhole  chord 
of  Ptolemy.  Another  improvement  on  Greek  trigonometry 
made  by  the  Arabs  points  likewise  to  Indian  influences: 
Operations  and  propositions  treated  by  the  Greeks  geometri- 
cally, are  expressed  by  the  Arabs  algebraically.  Thus  Al 

Battani  at  once  gets  from  an  equation =  D,  the  value  of 

COS0 

0  by  means  of  sin  0  =  D  -j-  VI  4- 1}2,  a  process  unknown  to 
Greek  antiquity.1  To  the  formulse  known  to  Ptolemy  he  adds 
an  important  one  of  his  own  for  oblique-angled  spherical  tri- 
angles ;  namely,  cos  a  —  cos  b  cos  c  +  sin  b  sin  c  cos  A. 

Important  are  the  researches  of  Abti'l  Wafd.  He  invented 
a  method  for  computing  tables  of  sines  which  gives  the  sine 
of  half  a  degree  correct  to  nine  decimal  places.2  He  and  Al- 
Battani,  in  the  study  of  shadow-triangles  cast  by  sun-dials, 
introduced  the  trigonometric  functions  tangent  and  cotangent; 
he  used  also  the  secant  and  cosecant.  An  important  change 
in  method  was  inaugurated  by  Dschdbir  ibn  Aflah  of  Seville 
in  Spain  (in  the  second  half  of  the  eleventh  century)  and  by 
Naslr  Eddln  (1201-1274)  in  distant  Persia.  In  the  works  of 
the  last  two  authors  we  find  for  the  first  time  trigonometry 
developed  as  a  part  of  pure  mathematics,  independently  of 
astronomy. 

1  CANTOR,  I.,  p.  737.  2  Consult  CANTOR,  I.,  746-748. 


EUROPE  DURING  THE  MIDDLE  AGES  131 

We  do  not  desire  to  go  into  greater  detail,  but  would  em- 
phasize the  fact  that  Nasir  Eddln  in  the  far  East,  during  a 
temporary  cessation  of  military  conquests  by  Tartar  rulers, 
developed  both  plane  and  spherical  trigonometry  to  a  very 
remarkable  degree.  Suter l  enthusiastically  asks,  what  would 
have  remained  for  European  scholars  of  the  fifteenth  century 
to  do  in  trigonometry,  had  they  known  of  these  researches  ? 
Or  were  some  of  them,  perhaps,  aware  of  these  investigations  ? 
To  this  question  we  can,  as  yet,  give  no  final  answer. 

Europe  during  the  Middle  Ages 

Introduction  of  Roman  Geometry.  —  Before  the  introduction 
of  Arabic  learning  into  Europe,  the  knowledge  of  geometry  in 
the  Occident  cannot  be  said  to  have  exceeded  that  of  the 
Egyptians  in  600  B.C.  The  monks  of  the  Middle  Ages  did  not 
go  much  beyond  the  definitions  of  the  triangle,  quadrangle, 
circle,  pyramid,  and  cone  (as  given  in  the  Eoman  encyclo- 
pedia of  the  Carthaginian,  Martianus  Capella),  and  the  simple 
rules  of  mensuration.  In  Alcum's  "  Problems  for  Quickening 
the  Mind,"  the  areas  of  triangular  and  quadrangular  pieces  of 
land  are  found  by  the  same  formulae  of  approximation  as 
those  used  by  the  Egyptians  and  given  by  Boethius  in  his 
geometry:  The  rectangle  equals  the  product  of  half  the  sums 
of  the  opposite  sides ;  the  triangle  equals  the  product  of  half 
the  sum  of  two  sides  and  half  the  third  side.  After  Alcuin, 
the  great  mathematical  light  of  Europe  was  Gerbert  (died 
1003).  In  Mantua  he  found  the  geometry  of  Boethius  and 
studied  it  zealously.  It  is  usually  believed  that  Gerbert  him- 
self was  the  author  of  a  geometry.  This  contains  nothing 

1  For  further  particulars  consult  his  article  in  Biblioth.  Mathem.,  1893, 
pp.  1-8. 


132  A   HISTORY   OF   MATHEMATICS 

more  than  the  geometry  of  Boethius,  but  the  fact  that  occa- 
sional errors  in  Boethius  are  herein  corrected  shows  that  the 

author  had  mastered  the  sub- 
^— ,  ject.1   "  The  earliest  mathemat- 

.  \   .  ical  paper  of  the  Middle  Ages 

which  deserves  this  name  is  a 


[/  \  letter  of  Gerbert  to  Adalbold, 


y~  \  bishop  of  Utrecht," 2  in  which 

is  explained  the   reason  why 


\| 
\| 


I  |     the  area  of  a  triangle,  obtained 

"  geometrically  "  by  taking  the 
product  of  the  base  by  its  altitude,  differs  from  the  area  calcu- 
lated "arithmetically,"  according  to  the  formula  -|-a(a  +  1)> 
used  by  surveyors,  where  a  stands  for  a  side  of  an  equilateral 
triangle.  Gerbert  gives  the  correct  explanation,  that  in  the 
latter  formula  all  the  small  rectangles,  into  which  the  triangle 
is  supposed  to  be  divided,  are  counted  in  wholly,  even  though 
parts  of  them  project  beyond  it. 

Translation  of  Arabic  Manuscripts.  —  The  beginning  of  the 
twelfth  century  was  one  of  great  intellectual  unrest.  Philos- 
ophers longed  to  know  more  of  Aristotle  than  could  be  learned 
through  the  writings  of  Boethius  ;  mathematicians  craved  a 
profounder  mathematical  knowledge.  Greek  texts  were  not 
at  hand;  so  the  Europeans  turned  to  the  Mohammedans  for 
instruction;  at  that  time  the  Arabs  were  the  great  scholars 
of  the  world.  We  read  of  an  English  monk,  Athelard  of 
Bath,  who  travelled  extensively  in  Asia  Minor,  Egypt,  and 
Spain,  braving  a  thousand  perils,  that  he  might  acquire  the 
language  and  science  of  the  .Mohammedans.  "  The  Moorish 

1  For  description  of  its  contents  see  CANTOR,  I.,  861-867  ;  S.  GUNTHER, 
Geschichte  des  Mathematischen  Unterrichts  im  deutchen  MittelaUer,  Ber- 
lin, 1887,  pp.  115-120. 

2  HANKEL,  p.  314. 


EUROPE  DURING  THE  MIDDLE   AGES  133 

universities  of  Cordova  and  Seville  and  Granada  were  dan- 
gerous resorts  for  Christians."  He  made  what  is  probably  the 
earliest  translation  from  the  Arabic  into  Latin  of  Euclid's 
Elements?*  in  1120.  He  translated  also  the  astronomical 
tables  of  Muhammed  ibn  Musa  Alchwarizmi.  In  his  trans- 
lation of  Euclid  from  the  Arabic  there  is  ground  for 
suspicion  that  Athelard  was  aided  by  a  previous  Latin 
translation.2 

All  important  Greek  mathematical  works  were  translated 
from  the  Arabic.  Gerard  of  Cremona  in  Lombardy  went  to 
Toledo  and  there  in  1175  translated  the  Almagest.  We  are 
told  that  he  translated  into  Latin  70  works,  embracing  the  15 
books  of  Euclid,  Euclid's  Data,  the  Sphcerica  of  Theodosius, 
and  a  work  of  Menelaus.  A  new  translation  of  Euclid's 

1  We  are  surprised  to  read  in  the  Dictionary  of  National  Biography 
[Leslie  Stephen's]  that  "it  has  not  yet  been  determined  whether  the 
translation  of  Euclid's  Elements  .  .  .  was  made  from  the  Arabic  version 
or  from  the  original."  To  our  knowledge  no  mathematical  historian  now 
doubts  that  ftie  translation  was  made  from  the  Arabic  or  suspects  that 
Athelard  used  the  Greek  text.  See  CANTOR,  I.,  713, 906 ;  II.,  100.  HANKEL, 
p.  335;  S.  GUNTHER,  Math.  Unt.  im  d.  Mittelalt.,  pp.  147-149;  W.  W. 
R.  BALL,  1893,  p.  170;  Gow,  p.  206;  H.  SUTER,  Gesch.  d.  Math.,  1., 
146  ;  HOEFER,  Histoire  des  Mathematiques,  1879,  p.  321.  Remarkable  is 
the  fact  that  MARIE,  in  his  12-volume  history  of  mathematics,  not  even 
mentions  Athelard.  He  says  that  Campanus  "a  donn6  des  Elements 
d'Euclide  la  premiere  traduction  qu'on  ait  eue  en  Europe."  MARIE,  II., 
p.  158. 

2  CANTOR,  II.,  100-102.  In  a  geometrical  manuscript  in  the  British 
Museum  it  is  said  that  geometry  was  invented  in  Egypt  by  Eucleides. 
This  verse  is  appended: 

"  Thys  craft  com  ynto  England,  as  y  ghow  say, 
Yn  tyme  of  good  Kyng  Adelstones  day." 

See  HALLIWELL'S  Kara  Mathematica,  London,  1841,  p.  56,  etc.  As  King 
Athelstan  lived  about  200  years  before  Athelard,  it  would  seem  that  a 
Latin  Euclid  (perhaps  only  the  fragments  given  by  Boethius)  was  known 
in  England  long  before  Athelard. 


134  A   HISTORY   OP   MATHEMATICS 

Elements  was  made,  about  1260,  by  Giovanni  Campano  (latin- 
ized form,  Campanus)  of  Novara  in  Italy.  It  displaced  the 
earlier  ones  and  formed  the  basis  of  the  printed  editions. 

The  First  Awakening. — The  central  figure  in  mathematical 
history  of  this  period  is  the  gifted  Leonardo  of  Pisa  (1175—?). 
His  main  researches  are  in  algebra,  but  his  Practica  Geometrice, 
published  in  1220,  is  a  work  disclosing  skill  and  geometric 
rigour.  The  writings  of  Euclid  and  of  some  other  Greek 
masters  were  known  to  him,  either  directly  from  Arabic  manu- 
scripts or  from  the  translations  made  by  his  countrymen, 
Gerard  of  Cremona  and  Plato  of  Tivoli.  Leonardo  gives  ele- 
gant demonstrations  of  the  "Heronic  Formula"  and  of  the 
theorem  that  the  medians  of  a  triangle  meet  in  a  point  (known 
to  Archimedes,  but  not  proved  by  him).  He  also  gives  the 
theorem  that  the  square  of  the  diagonal  of  a  rectangular 
parallelepiped  is  equal  to  the  sum  of  the  three  squares  of  its 
sides.1  Algebraically  are  solved  problems  like  this:  To  in- 
scribe in  an  equilateral  triangle  a  square  resting  upon  the  base 
of  the  triangle. 

A  geometrical  work  similar  to  Leonardo's  in  Italy  was 
brought  out  in  Germany  about  the  same  time  by  the  monk 
Jordanus  Nemorarius.  It  was  entitled  De  triangulis,  and  was 
printed  by  Curtze  in  1887.  It  indicates  a  decided  departure 
from  Greek  models,  though  to  Euclid  reference  is  frequently 
made.  There  is  nothing  to  show  that  it  was  used  anywhere  as  a 
text-book  in  schools.  This  work,  like  Leonardo's,  was  probably 
read  only  by  the  elite.  As  specimens  of  remarkable  theorems, 
we  give  the  following :  If  circles  can  be  inscribed,  and  circum- 
scribed about  an  irregular  polygon,  then  their  centres  do  not 
coincide ;  of  all  inscribed  triangles  having  a  common  base,  the 
isosceles  is  the  maximum.  Jordanus  accomplishes  the  trisec- 
tion  of  an  angle  by  giving  a  graduated  ruler  simultaneously  a 
i  CANTOR,  II.,  p.  39. 


EUROPE   DURING    THE   MIDDLE   AGES  135 

rotating  and  a  sliding  motion,  its  final  position  being  fixed 
with  aid  of  a  certain  length  marked  on  the  ruler.1  In  this 
trisection  he  does  not  permit  himself  to  be  limited  to  Euclid's 
postulates,  which  allow  the  use  simply  of  an  unmarked  ruler 
and  a  pair  of  compasses.  He  also  introduces  motion  of  parts  of 
a  figure  after  the  manner  of  some  Arabic  authors.  Such  motion 
is  foreign  to  Euclid's  practice.2  The  same  mode  of  trisection 
was  given  by  Campanus. 

Jordanus's  attempted  exact  quadrature  of  the  circle  lowers 
him  in  our  estimation.  Circle-squaring  now  began  to  com- 
mand the  lively  attention  of  mathematicians.  Their  efforts 
remained  as  futile  as  though  they  had  attempted  to  jump  into 
a  rainbow;  the  moment  they  thought  they  had  touched  the 
goal,  it  vanished  as  by  magic,  and  was  as  far  as  ever  from 
their  reach.  In  their  excitement  many  of  them  became  sub- 
ject to  mental  illusions,  and  imagined  that  they  had  actually 
attained  their  aim,  and  were  in  the  midst  of  a  triumphal  arch 
of  glory,  the  wonder  and  admiration  of  the  world. 

The  fourteenth  and  fifteenth  centuries  have  brought  forth 
no  geometricians  who  equalled  Leonardo  of  Pisa.  Much  was 
written  on  mathematics,  and  an  effort  put  forth  to  digest  the 
rich  material  acquired  from  the  Arabs.  No  substantial  con- 
tributions were  made  to  geometry. 

An  English  manuscript  of  the  fourteenth  century,  on  sur- 
veying, bears  the  title :  Nowe  sues  here  a  Tretis  of  Geometri 
wherby  you  may  Jen  owe  the  heghte,  depnes,  and  the  brede  of  most- 
wheat  erthely  thynges.3  The  oldest  French  geometrical  manu- 
script (of  about  1275)  is  likewise  anonymous.  Like  the  Eng- 
lish treatise,  it  deals  with  mensuration.  From  the  study  of 

1  CANTOR,  II.,  81,  gives  the  construction  in  full. 

2  For  fuller  extracts  from  the  De  triangulis,  see  CANTOR,  II. ,  73-79; 
S.  GUNTHER,  op.  cit.,  160-162. 

8  See  HALLIWELL,  Rara  Mathematica,  56-71 ;  CANTOB,  II.,  p.  112. 


136  A    HISTORY   OF   MATHEMATICS 

manuscripts,  made  to  the  present  time,  it  would  seem  that 
since  the  thirteenth  century  surveying  in  Europe  had  de- 
parted from  B-oman  models  and  come  completely  under  the 
influence  of  the  Greece-Arabic  writers.1  An  author  of  consid- 
erable prominence  was  Thomas  Bradwardine  (1290  ?-1349), 
archbishop  of  Canterbury.  He  was  educated  at  Merton  Col- 
lege, Oxford,  and  later  lectured  in  that  university  on  theology, 
philosophy,  and  mathematics.  His  philosophic  writings  con- 
tain able  discussions  of  the  infinite  and  the  infinitesimal  —  sub- 
jects which  thenceforth  came  to  be  studied  in  connection  with 
mathematics.  Bradwardine  wrote  several  mathematical  treat- 
ises. A  Geometria  speculative!,  was  printed  in  Paris  in  1511  as 
the  work  of  Bradwardinus,  but  has  been  attributed  by  some  to 
a  Dane,  named  Petrus,  then  a  resident  of  Paris.  This  remarka- 
ble work  enjoyed  a  wide  popularity.  It 
treats  of  the  regular  solids,  of  isoperimetrical 
figures  in  the  manner  of  Zenodorus,  and  of 
star-polygons.  The  first  appearance  of  such 
polygons  was  with  Pythagoras  and  his  school. 
The  pentagram-star  was  used  by  the  Pythag- 
oreans as  a  badge  or  symbol  of  recognition,  and  was  called 
by  them  Health.2  We  next  meet  such  polygons  in  the  geom- 
etry of  Boethius,  in  the  translation  of  Euclid  from  the 
Arabic  by  Athelard  of  Bath,  and  by  Campanus,  and  in  the 
earliest  French  geometric  treatise,  mentioned  above.  Brad- 
wardine develops  some  geometric  properties  of  star-polygons 
—  their  construction  and  angle-sum.  We  encounter  these  fas- 
cinating figures  again  in  Begiomontanus,  Kepler,  and  others. 
In  Bradwardine  and  a  few  other  British  scholars  England 
proudly  claims  the  earliest  European  writers  on  trigonome- 
try. Their  writings  contain  trigonometry  drawn  from  Arabic 

i  CANTOR,  II.,  127.  2  Gow,  p.  151. 


A 


EUROPE   DURING   THE   MIDDLE    AGES  137 

sources.  John  Maudith,  professor  at  Oxford  about  1340, 
speaks  of  the  umbra  ("  tangent ") ;  Bradwardine  uses  the 
terms  umbra  recta  ("  cotangent ")  and  umbra  versa  ("  tangent  ")* 
which  occur  in  Gerard  of  Cremona's  translation  of  Al-Arzakel's 
Toledian  tables.  The  Hindus  had  introduced  the  sine,  versed 
sine,  cosine;  the  Arabs  the  tangent,  cotangent,  secant,  cosecant. 

Perhaps  the  greatest  result  of  the  introduction  of  Arabic 
learning  was  the  establishment  of  universities.  What  was 
their  attitude  toward  mathematics?  At  the  University  of 
Paris  geometry  was  neglected.  In  1336  a  rule  was  intro- 
duced that  no  student  should  take  a  degree  without  attending 
lectures  on  mathematics,  and  from  a  commentary  on  the  first 
six  books  of  Euclid,  dated  1536,  it  appears  that  candidates 
for  the  degree  of  A.M.  had  to  take  oath  that  they  had 
attended  lectures  on  these  books.2  Examinations,  when  held 
at  all,  probably  did  not  extend  beyond  the  first  book,  as  is 
shown  by  the  nickname  "  magister  matheseos  "  applied  to  the 
theorem  of  Pythagoras,  the  last  of  the  book.  At  Prague, 
founded  in  1384,  astronomy  and  applied  mathematics  were 
additional  requirements.  Roger  Bacon,  writing  near  the  close 
of  the  thirteenth  century,  says  that  at  Oxford  there  were  few 
students  who  cared  to  go  beyond  the  first  three  or  four  propo- 
sitions of  Euclid,  and  that  on  this  account  the  fifth  proposition 
was  called  "elefuga,"  that  is,  "flight  of  the  wretched."  We 
are  told  that  this  fifth  proposition  was  later  called  the  "  pons 
asinorum  "  or  "  the  Bridge  of  Asses." 3  Clavius  in  his  Euclid, 
edition  of  1591,  says  of  this  theorem,  that  beginners  find  it 

1  CANTOR,  II.,  111. 

2  HANKEL,  pp.  354-359.    We  have  consulted  also  H.  SUTER,  Die  Mathe- 
matik  auf  den  Universitaten  des  Mittelalters,  Zurich,  1887  ;  S.  GUNTHER, 
Math.  Unt.  im  d.  Mittela.,  p.  199 ;  CANTOR,  II.,  pp.  139-142. 

8  This  nickname  is  sometimes  also  given  to  the  Pythagorean  Theorem, 
I.,  47,  though  usually  I.,  47,  is  called  "the  windmill."  Read  THOMAS 
CAMPBELL'S  poem,  '*  The  Pons  Asinorum." 


138  A   HISTORY   OF  MATHEMATICS 

difficult  and  obscure,  on  account  of  the  multitude  of  lines 
and  angles,  to  which  they  are  not  yet  accustomed.  These 
last  words,  no  doubt,  indicate  the  reason  why  geometrical 
study  seems  to  have  been  so  pitifully  barren.  Students  with 
no  kind  of  mathematical  training,  perhaps  unable  to  perform 
the  simplest  arithmetical  computations,  began  to  memorize 
the  abstract  definitions  and  propositions  of  Euclid.  Poor 
preparation  and  poor  teaching,  combined  with  an  absence  of 
rigorous  requirements  for  degrees,  probably  explain  this  flight 
from  geometry  —  this  "elefuga."  In  the  middle  of  the  fif- 
teenth century  the  first  two  books  were  read  at  Oxford. 

Thus  it  is  seen  that  the  study  of  mathematics  was  main- 
tained at  the  universities  only  in  a  half-hearted  manner. 


MODERN   TIMES 

ARITHMETIC 
Its  Development  as  a  Science  and  Art 

DURING  the  sixteenth  century  the  human  mind  made  an 
extraordinary  effort  to  achieve  its  freedom  from  scholastic  and 
ecclesiastical  bondage.  This  independent  and  vigorous  intel- 
lectual activity  is  reflected  in  the  mathematical  books  of  the 
time.  The  best  arithmetical  work  of  the  fifteenth  as  also 
of  the  sixteenth  century  emanated  from  Italian  writers, — 
Lucas  Pacioli  and  Tartaglia.  Lucas  Pacioli  (1445  ?-1514  ?) 
—  also  called  Lucas  di  Burgo,  Luca  Paciuolo,  or  Pacciuolus— 
was  a  Tuscan  monk  who  taught  mathematics  at  Perugia, 
Naples,  Milan,  Florence,  Rome,  and  Venice.  His  treatise, 
Summa  de  Arithmetica,  1494,  contains  all  the  knowledge  of 
his  day  on  arithmetic,  algebra,  and  trigonometry,  and  is 
the  first  comprehensive  work  which  appeared  after  the 
liber  abaci  of  Fibonaci.  The  earliest  arithmetic  known  to  have 
been  brought  out  in  print  anywhere  appeared  anonymously  in 
Treviso,  Italy,  in  1478,  and  is  known  as  the  "  Treviso  arithmetic." 

Tartaglia's  real  name  was  Nicolo  Fontana  (1499  ?-1557). 
When  a  boy  of  six,  Nicolo  was  so  badly  cut  by  a  French 
soldier,  that  he  never  again  gained  the  free  use  of  his  tongue. 
Hence  he  was  called  Tartaglia,  i.e.  the  stammerer.  His 
widowed  mother  being  too  poor  to  pay  his  tuition  at  school,  he 

139 


140  A   HISTORY   OF   MATHEMATICS 

learned  to  read  and  acquired  a  knowledge  of  Latin,  Greekt 
and  mathematics  without  a  teacher.  Possessing  a  mind  of 
extraordinary  power,  he  was  able  to  teach  mathematics  at  an 
early  age.  He  taught  at  Verona,  Piacenza,  Venice,  and 
Brescia.  It  was  his  intention  to  embody  his  original  re- 
searches in  a  great  work,  General  trattato  di  numeri  et  misure, 
but  at  his  death  it  was  still  unfinished.  The  first  two  parts 
were  published  in  1556,  and  treat  of  arithmetic.  Tartaglia 
discusses  commercial  arithmetic  somewhat  after  the  manner 
of  Pacioli,  but  with  greater  fulness  and  with  simpler  and 
more  methodical  treatment.  His  work  contains  a  large  num- 
ber of  exercises  and  problems  so  arranged  as  to  insure  the 
reader's  mastery  of  one  subject  before  proceeding  to  the  next. 
Tartaglia  bears  constantly  in  mind  the  needs  of  the  practical 
man.  His  description  of  numerical  operations  embraces  seven 
different  modes  of  multiplication  and  three  methods  of  divis- 
ion.1 He  gives  the  Venetian  weights  and  measures. 

Mathematical  study  was  fostered  in  Germany  at  the  close 
of  the  fifteenth  century  by  Georg  Purbach  and  his  pupil, 
Regiomontanus.  The  first  printed  German  arithmetic  appeared 
in  1482  at  Bamberg.  It  is  by  Ulrich  Wagner,  a  practitioner  of 
Niirnberg.  It  was  printed  on  parchment,  but  only  fragments 
of  one  copy  are  now  extant.2  In  1483  the  same  Bamberg 
publishers  brought  out  a  second  arithmetic,  printed  on  paper, 
and  covering  77  pages.  The  work  is  anonymous,  but  Ulrich 
Wagner  is  believed  to  be  its  author.  It  is  worthy  of  remark 
that  the  earliest  printed  German  arithmetic  appeared  only  four 
years  after  the  first  printed  Italian  arithmetic.  The  Bam- 
berg arithmetic  of  1483,  says  linger,  bears  no  resemblance 
to  previous  Latin  treatises,  but  is  purely  commercial.  Modelled 
after  it  is  the  arithmetic  by  John  Widmann,  Leipzig,  1489. 

1  UNGER,  p.  60.  2  UNGBR,  pp.  36-40. 


MODERN   TIMES  141 

This  work  has  become  famous,  as  being  the  earliest  book  in 
which  the  symbols  +  and  —  have  been  found.  They  occur  in 
connection  with  problems  worked  by  "  false  position."  Wid- 
mann  says,  "what  is  — ,  that  is  minus;  what  is  +,  that  is 
more."  The  words  "minus"  and  "more,"  or  "plus,"  occur 
long  before  Widmann's  time  in  the  works  of  Leonardo  of 
Pisa,  who  uses  them  in  connection  with  the  method  of 
false  position  in  the  sense  of  "  positive  error "  and  "  negative 
error."  *  While  Leonardo  uses  "  minus  "  also  to  indicate  an 
operation  (of  subtraction),  he  does  not  so  use  the  word  "  plus." 
Thus,  7  -f-  4  is  written  "  septem  et  quatuor."  The  word 
"plus,"  signifying  the  operation  of  addition,  was  first  found 
by  Enestrom  in  an  Italian  algebra  of  the  fourteenth  century. 
The  words  "  plus  "  and  "  minus,"  or  their  equivalents  in  the 
modern  tongues,  were  used  by  Pacioli,  Chuquet,  and  Widmann. 
It  is  proved  that  the  sign  -{-  comes  from  the  Latin  et,  as  it  was 
cursively  written  at  the  time  of  the  invention  of  printing. 
The  origin  of  the  sign  —  is  still  uncertain.  These  signs  were 
used  in  Italy  by  Leonardo  da  Vinci  very  soon  after  their  appear- 
ance in  Widmann's  work.  They  were  employed  by  Grammateus 
(Heinrich  Schreiber),  a  teacher  at  the  University  of  Vienna,  by 
Christoff  Kudolff  in  his  algebra,  1525,  and  by  Stifel  in  1553. 
Thus,  by  slow  degrees,  their  adoption  became  universal. 

During  the  early  half  of  the  sixteenth  century  some  of 
the  most  prominent  German  mathematicians  (Grammateus, 
Rudolff,  Apian,  Stifel)  contributed  toward  the  preparation  of 

1  G.  ENESTROM  in  U Intermediate  des  Mathematiciens,  1894,  pp.  119- 
120.  Regarding  the  supposed  origin  of  +  and  —  consult  also  ENESTROM 
in  Ofversigt  af  Kongl.  Vetenskaps  —  Akademiens  Forhandlingar,  Stock- 
holm, 1894,  pp.  243-256;  CANTOR,  II.,  211-212;  DE  MORGAN  in  Philo- 
sophical Magazine,  20,  1842,  pp.  135-137 ;  in  Trans,  of  the  Philos. 
Soc.  of  Cambridge,  11,  p.  203 ;  Enestrom,  Biblioth.  Math.  (3)  9,  p.  165  ; 
10,  p.  182. 


142  A   HISTORY   OF  MATHEMATICS 

• 

practical  arithmetics,  but  after  that  period  this  important 
work  fell  into  the  hands  of  the  practitioners  alone.1  The  most 
popular  of  the  early  text-book  writers  was  Adam  Miese,  who 
published  several  arithmetics,  but  that  of  1522  is  the  one 
usually  associated  with  his  name. 

A  French  work  which  in  point  of  merit  ranks  with  Pacioli's 
Summa  de  Arithmetica,  but  which  was  never  printed  before  the 
nineteenth  century,  is  Le  Triparty  en  la  science  des  nombres, 
written  in  1484  in  Lyons  by  Nicolas  Chuquet.2  A  contempo- 
rary of  Chuquet,  in  France,  was  Jacques  Lefevre,  who  brought 
out  printed  editions  of  older  mathematical  works.  For  instance, 
in  1496  there  appeared  in  print  the  arithmetic  of  the  German 
monk,  Jordanus  Nemorarius,  a  work  modelled  after  the  arith- 
metic of  Boethius,  and  at  this  time  over  two  centuries  old. 
A  quarter  of  a  century  later,  in  1520,  appeared  a  popular 
French  arithmetic  by  Estienne  de  la  Roche,  named  also 
Villefranche.  The  author  draws  his  material  mainly  from 
Chuquet  and  Pacioli.3 

We  proceed  now  to  the  discussion  of  a  few  arithmetical 
topics.  Down  to  the  seventeenth  century  great  diversity  and 
clumsiness  prevailed  in  the  numeration  of  large  numbers. 
Italian  authors  grouped  digits  into  periods  of  six,  others 
sometimes  into  periods  of  three.  Adam  Biese,  who  did  more 
than  any  one  else  in  the  first  half  of  the  sixteenth  century 
toward  spreading  a  knowledge  of  arithmetic  in  Germany, 

writes  S6  78§  325  178,  and  reads,  "  Sechs  und  achtzig 
tausend,  tausend  mal  tausend,  sieben  hundert  tausend  mal 
tausendt,  neun  vnnd  achtzig  tausend  mal  tausend,  drei  hun- 
dert tausent,  filnff  vnnd  zwantzig  tausend,  ein  hundert,  acht 

1  UNGER,  p.  44. 

2  The  Triparty  is  printed  in  Bulletin  Boncompagni,  XIII.,  r>85-592. 
A  description  of  the  work  is  given  by  CANTOR,  II.,  347-355. 

8  CANTOR,  II.,  371. 


MODERN   TIMES  143 

undsiebentzig."1  Stif el  in  1544  writes  2  329  089  562  800, 
and  reads,  "  duo  millia  millies  millies  millies  -,  trecenta  viginti 
novem  millia  millies  millies ;  octoginta  novem  millia  millies ; 
quingenta  sexaginta  duo  millia;  octingenta."  Tonstall,  in 
1522,  calls  109  "millies  millena  millia."2  This  habit  of 
grouping  digits,  for  purposes  of  numeration,  did  not  exist 
among  the  Hindus.  They  had  a  distinct  name  for  each  suc- 
cessive step  in  the  scale,  and  it  has  been  remarked  that 
this  fact  probably  helped  to  suggest  to  them  the  principle  of 
local  value.  They  read  86789325178  as  follows :  "  8  kharva, 
6  padma,  7  vyarbuda,  8  koti,  9  prayuta,  3  laksha,  2  ayuta, 
5  sahasra,  1  qata,  7  daqon,  8."3  One  great  objection  to  this 
Hindu  scheme  is  that  it  burdens  the  memory  with  too  many 
names. 

The  first  improvement  on  ancient  and  mediaeval  methods 
of  numeration  was  the  invention  of  the  word  millione  by  the 
Italians  in  the  fourteenth  century,  to  signify  great  thousand* 
or  10002.  This  new  word  seems  originally  to  have  indicated 
a  concrete  measure,  10  barrels  of  gold.5  The  words  millione, 
nulla  or  cero  (zero)  occur  for  the  first  time  in  print  in  the 
Arithmetica  of  Borgi  (1484).  In  the  next  two  centuries  the  use 
of  millione  spread  to  other  European  countries.  Tonstall,  in 
1522,  speaks  of  the  term  as  common  in  England,  but  rejects 
it  as  barbarous !  The  seventh  place  in  numeration  he  calls 
"millena  millia;  vulgus  millionem  barbare  vocat."7  Dticange 
of  Rymer  mentions  the  word  million  in  1514 ; 8  in  1540  it  occurs 
once  in  the  arithmetic  of  Christoff  Rudolff. 

The  next  decided  advance  was  the  introduction  of  the  words 

1  WILDE RMUTH,   article  "Rechnen"  in  Encyklopcedie  des  gesammten 
Erziehungs- und  Unterrichtswesens,  DR.  K.  A.  SCHMID,  1885,  p.  794. 

2  PEACOCK,  p.  426.  6  HANKEL,  p.  14.  7  PEACOCK,  p.  426. 
8  HANKEL,  p.  15.                 •  CANTOS,  II.,  305.             8  WILDERMUTH. 

*  PEACOCK,  p.  378. 


144  A   HISTORY   OF   MATHEMATICS 

billion,  trillion,  etc.  Their  origin  dates  back  almost  to  the 
time  when  the  word  million  was  first  used.  So  far  as  known, 
they  first  occur  in  a  manuscript  work  on  arithmetic  by  that 
gifted  French  physician  of  Lyons,  Nicolas  Chuquet.  He  em- 
ploys the  words  byllion,  tryllion,  quadrillion,  quyllion,  sixlion, 
septyllion,  octyllion,  nonyllion,  "et  ainsi  des  aultres  se  plus 
oultre  on  voulait  proceder,"  to  denote  the  second,  third,  etc. 
powers  of  a  million,  i.e.  (1000,000)2,  (1000,000)3,  etc.1  Evi- 
dently Chuquet  had  solved  the  difficult  question  of  numera- 
tion. The  new  words  used  by  him  appear  in  1520  in  the 
printed  work  of  La  Roche.  Thus  the  great  honour  of  having 
simplified  numeration  of  large  numbers  appears  to  belong  to 
the  French.  In  England  and  Germany  the  new  nomenclat- 
ure was  not  introduced  until  about  a  century  and  a  half 
later.  In  England  the  words  billion,  trillion,  etc.,  were  new 
when  Locke  wrote,  about  1687. 2  In  Germany  these  new  terms 
appear  for  the  first  time  in  1681  in  a  work  by  Heckenberg  of 
Hanover,  but  they  did  not  come  into  general  use  before  the 
eighteenth  century.3 

About  the  middle  of  the  seventeenth  century  it  became  the 
custom  in  France  to  divide  numbers  into  periods  of  three 
digits,  instead  of  six,  and  to  assign  to  the  word  billion,  in 
place  of  the  old  meaning,  (1000,000)2  or  1C12,  the  new  meaning 
of  109.4  The  words  trillion,  quadrillion,  etc.,  receive  the  new 
definitions  of  1012,  1015,  etc.  At  the  present  time  the  words 
billion,  trillion,  etc.,  mean  in  France,  in  other  south-European 
countries,  and  in  the  United  States  (since  the  first  quar- 

1  CANTOR,  II.,  348. 

2  LOCKE,  Human  Understanding,  Chap.  XVI. 
8  UNGER,  p.  71. 

4  Dictionnaire  de  la  Langue  Franqaise  par  E.  LITTRE.  It  is  interest- 
ing to  notice  that  Bishop  Berkeley,  when  a  youth  of  twenty-three,  pub- 
lished in  Latin  an  arithmetic  (1707),  giving  the  words  billion,  trillion, 
etc. ,  with  their  new  meanings. 


MODERN   TIMES  145 

ter  of  this  century),  109,  1012,  etc. ;  while  in  Germany,  Eng- 
land, and  other  north-European  countries  they  mean  1012, 
1018,  etc. 

From  what  has  been  said  it  appears  that,  while  the  Arabic 
notation  of  integral  numbers  was  brought  to  perfection  by  the 
Hindus  as  early  as  the  ninth  century,  our  present  numeration 
dates  from  the  close  of  the  fifteenth  century.  One  of  the 
advantages  of  the  Arabic  notation  is  its  independence  of  its 
numeration.  At  present  the  numeration,  though  practically 
adequate,  is  not  fully  developed.  To  read  a  number  of,  say, 
1000  digits,  or  to  read  the  value  of  TT,  calculated  to  707 
decimal  places  by  William  Shanks,  we  should  have  to  invent 
new  words. 

A  good  numeration,  accompanied  by  a  good  notation,  is 
essential  for  proficient  work  in  numbers.  We  are  told  that 
the  Yancos  on  the  Amazon  could  not  get  beyond  the  number 
three,  because  they  could  not  express  that  idea  by  any  phrase- 
ology more  simple  than  Poettarrarorincoaroae.* 

As  regards  arithmetical  operations,  it  is  of  interest  to  notice 
that  the  Hindu  custom  was  introduced  into  Europe,  of  begin- 
ning an  addition  or  subtraction,  sometimes  from  the  right, 
but  more  commonly  from  the  left.  Notwithstanding  the  in- 
convenience of  this  latter  procedure,  it  is  found  in  Europe 
as  late  as  the  end  of  the  sixteenth  century.2 

Like  the  Hindus,  the  Italians  used  many  different  methods 
in  the  multiplication  of  numbers.  An  extraordinary  passion 
seems  to  have  existed  among  Italian  practitioners  of  arith- 
metic, at  this  time,  for  inventing  new  forms.  Pacioli  and 
Tartaglia  speak  slightingly  of  these  efforts.3  Pacioli  himself 
gives  eight  methods  and  illustrates  the  first,  named  bericuocoli 
or  scacherii,  by  the  first  example  given  here  : 4 

1  PEACOCK,  p.  390.        3  PEACOCK,  p.  431. 

2  PEACOCK,  p.  427.        4  PEACOCK,  p.  429  ;  CANTOR,  II.,  311. 

L 


146 


A    HISTORY   OF   MATHEMATICS 


9876 

6789 

•|8|8|8|8|4| 
|7|9|0|0|8| 
|6|9|1|3|2| 
|5|9|2|5|6| 


9876 
6789 


61101000 

5431200 

475230 

40734 


67048164  67048164 

The  second  method,  for  some  unknown  reason  called  castel- 
i.e.  "by  the  little  castle/'  is  shown  here  in  the  second 
example.  Tho  third  method  (not  illustrated  here)  invokes 
the  aid  of  tables;  the  fourth,  crocetta  sine  casella,  i.e.  "by 
cross  multiplication,"  though  harder  than  the  others,  was 
practised  by  the  Hindus  (named  by  them  the  "lightning" 
method),  and  greatly  admired  by  Pacioli.  See  our  third 
example,  in  which  the  product  is  built  up  as  follows : 

3  •  6  + 10  (3  •  1  +  5  .  6)  +  100  (3  -  4  +  5  . 1  +  2  .  6) 

+ 1000  (5.4  +  2-1)  + 10,000  (2  •  4). 

In  Pacioli's  fifth  method,  quadrilatero,  or  "  by  the  square,"  the 
digits  are  entered  in  a  square  divided  up 

A.  1  £: 

like  a  chessboard  in  Hindu  fashion,  and 
are  added  diagonally.     His  sixth  is. called 
gelosia  or   graticola,   i.e.    "latticed    mul- 
tiplication"   (see    our    fourth    example, 
987  x  987).      It  is  so  named  because  the  figure  looks  like  a 
lattice   or   grating,   such    as   was    then    placed    in    Venetian 
987  windows  that  ladies  and  nuns  might 

not  easily  be  seen  from  the  street.1 
The  word  gelosia  means  primarily 
jealousy.  The  last  two  of  Pacioli's 
methods  are  illustrated,  respectively, 
by  the  examples  234x48=234x6x8 
and  163  x  17  =  163  x  10  +  163  x  7. 
1  PEACOCK,  p.  431. 


1O5248 


MODERN  TIMES  147 

Some  books  also  give  the  mode  of  multiplication  devised  by  the 
early  Arabs  in  imitation  of  one  of  the  Hindu  methods.  We 
illustrate  it  in  the  example 1  359  x  837  =  300483.  It  would  be 
erroneous  to  conclude  from  the  existence  of  these  and  other 
methods  that  all  of  them  were  in  actual  use.  As  a  matter 
of  fact  the  first  of  Pacioli's  methods  (now  in  common  use) 
was  the  one  then  practised  almost  exclusively. 
The  second  method  of  Pacioli,  illustrated  in  our  0  Q 

second  exam  pie,  would  have  been  a  better  choice,         0798 
as  we  expect  to  show  later.  9672 

It  is  worthy  of  remark  that  the  Hindus  and  |  8  7  2  7  3 
Arabs  apparently  did  not  possess  a  multipli-  35999 
cation  table,  such,  for  instance,  as  the  one  given  355 

by  Boethius,2  which  was  arranged  in  a  square. 
We  show  it  here  as  far  as  4  x  4.     The  Italians        1234 
gave  this  table  in  their  arithmetics.      Another        2468 
form  of   it.  the  triangular,  shown  here  as  far        3  6    9  12 

4    o    -I  o    -I/* 

as    4x4,    sometimes    occurs    in    arithmetical 

books.      Some    writers    (for    instance,    Finaeus  i 

in  France  and  Recorde  in  England)  teach  also  2    * 

q       a        Q 

a    kind   of    complementary    multiplication,    re- 

4     o     \2t   lo 

sembling  a  process  first  found  among  the 
Romans.  It  is  frequently  called  the  "sluggard's  rule,"  and 
was  intended  to  relieve  the  me'mory  of  all  products  of  digits 
exceeding  5.  It  is  analogous  to  the  process  in  Gerbert's  comple- 
mentary division :  "  Subtract  each  digit  from  10,  and  write  down 
the  differences  together,  and  add  as  many  tens  to  their 

7       Q 

X     product  as  the   \  first      I    digit  exceeds  the   j  second  I 
g     4  (second)  C  first 

difference."3    If  a  and  b  designate  the  digits,  then  the 
rule  rests  on  the  identity  (10  -  a)  (10  -  6) + 10  (a + b  - 10) = ab. 

1  UNGER,  p.  77.  2  BOETHIUS  (FRIEDLEIN'S  Ed.),  P-  63. 

8  PEACOCK,  p.  432. 


148  A   HISTORY   OF   MATHEMATICS 

Division  was  always  considered  an  operation  of  considerable 
difficulty.  Pacioli  gives  four  methods.  The  first,  "  division 
by  the  head,"  is  used  when  the  divisor  consists  of  one  digit  or 
two  digits  (such  as  12,  13),  included  in  the  Italian  tables 
of  multiplication.1  In  the  second,  division  is  performed 
successively  by  the  simple  factors  of  the  divisor.  The  third 
method,  "by  giving,"  is  so  called  because  after  each  subtrac- 
tion we  give  or  add  one  more  figure  on  the  right  hand.  This 
is  the  method  of  "  long  division  "  now  prevalent.  But  Pacioli 
expended  his  enthusiasm  on  the  fourth  method,  called  by  the 
Italians  the  "  galley,"  because  the  digits  in  the  completed 
work  were  arranged  in  the  form  of  that  vessel.  He  considered 
this  procedure  the  swiftest,  just  as  the  galley  was  the  swiftest 
ship.  The  English  call  it  the  scratch  method.  The  complete 
division  of  59078  by  74  is  shown  in  Fig.  I.2 

02  6 

V  W 

n         in 

59078(         ^078(7         ^078(7 
74  ft  ^4 

m  7  jr 

FIG.  1.  FIG.  2.  FIG.  8.  FIG.  4.  FIG.  5. 

The  other  four  figures  show  the  division  in  its  successive 
stages.3 

1  PEACOCK,  p.  432. 

2  This  example  is  taken  from  the  early  German  mathematician,  Pur- 
bach,  by  ARNO  SADOWSKI,  Die  osterreichiche  Rechenmethode,  Konigsberg, 
1892,  p.  14.     Pacioli's  illustration  of  the  galley  method  is  given  by 
PEACOCK,  p.  433,  UNGER,  p.  79. 

8  The  work  proceeds  as  follows  :  Fig.  2  shows  the  dividend  and  divisor 
written  in  their  proper  positions,  also  a  curve  to  indicate  the  place  for  the 
quotient.  Write  7  in  the  quotient ;  then  7  x  7  =  49,  59  —  49  =  10  ;  write 
10  above,  scratch  the  59  and  also  the  7  in  the  divisor.  7  x  4  =  28  ;  the  4 


MODERN   TIMES  149 

For  a  long  time  the  galley  or  scratch  method  was  used  almost 
to  the  entire  exclusion  of  the  other  methods.  As  late  as  the 
seventeenth  century  it  was  preferred  to  the  one  now  in  vogue. 
It  was  adopted  in  Spain,  Germany,  and  England.  It  is  found 
in  the  works  of  Tonstall,  Recorde,  Stifel,  Stevin,  Wallis, 
Napier,  and  Oughtred.  Not  until  the  beginning  of  the 
eighteenth  century  was  it  superseded  in  England.1  It  will 
be  remembered  that  the  scratch  method  did  not  spring  into 
existence  in  the  form  taught  by  the  writers  of  the  sixteenth 
century.  On  the  contrary,  it  is  simply  the  graphical  repre- 
sentation of  the  method  employed  by  the  Hindus,  who  calcu- 
lated with  a  coarse  pencil  on  a  small  dust-covered  tablet.  The 
erasing  of  a  figure  by  the  Hindus  is  here  represented  by  the 
scratching  of  a  figure.  On  the  Hindu  tablet,  our  example, 
taken  from  Purbach,  would  have  appeared,  when  completed, 
as  follows :  26 


59078 


798 


74 

The  practice  of  European  arithmeticians  to  prove  their 
operations  by  "  casting  out  the  9's  "  was  another  method,  useful 
to  the  Hindus,  but  poorly  adapted  for  computation  on  paper 
or  slate,  since  in  this  case  the  entire  operation  is  exhibited  at 
the  close,  and  all  the  steps  can  easily  be  re-examined. 

being  under  the  0  in  the  dividend,  28  must  be  subtracted  from  100  ;  the 
remainder  is  72  ;  scratch  the  10,  the  0  of  the  dividend,  and  the  4  of  the 
divisor  (Fig.  3)  ;  above  write  the  7  and  2.  Write  down  the  divisor  one 
place  further  to  the  right,  as  in  Fig.  4.  Now  7  into  72  goes  9  times  ; 
9  x  7  =  63  ;  72  -  63  =  9  ;  scratch  72  and  the  7  below,  write  9  above  ; 
9  x  4  =  36  ;  97  -  36  =  61 ;  scratch  the  9  above,  write  6  above  it ;  scratch 
7  in  the  dividend  and  write  1  above  it ;  scratch  7  and  4  below,  Fig.  5.  Again 
move  the  divisor  one  place  to  the  right.  It  goes  8  times ;  7  x  8  =  56 ; 
61  —  56  =  5 ;  scratch  6  and  1  above  and  write  5  above  1 ;  8  x  4  =  32  ; 
58  —  32  =  26  ;  scratch  the  5  above  and  write  2  ;  above  the  scratched  8  in 
the  dividend  write  6,  Fig.  1.  The  remainder  is  26. 
1  PEACOCK,  p.  434. 


150  A   HISTORY  OF  MATHEMATICS 

As  there  were  two  rival  methods  of  division  ("  by  giving " 
and  "  galley  "),  there  were  also  two  varieties  in  the  extraction 
of  square  and  cube  root.1  In  case  of  surds,  great  interest  was 
taken  in  the  discovery  of  rules  of  approximation.  Leonardo 
of  Pisa,  Tartaglia,  and  others  give  the  Arabic  rule  (found,  for 
instance,  in  the  works  of  the  Arabs,  Ibn  Albanna  and  Alkal- 
sadi),  which  may  be  expressed  in  our  algebraic  symbols  thus : 2 


This  yields  the  root  in  excess,  while  the  following  Arabic  rule 
makes  it  too  small : 


Similar  formulae  were  devised  for  cube  root. 

In  other  methods  of  approximation  to  the  roots  of  surds,  the 
idea  of  decimal  fractions  makes  its  first  appearance,  though 
their  true  nature  and  importance  were  overlooked.  About  the 
middle  of  the  twelfth  century,  John  of  Seville,  presumably  in 
imitation  of  Hindu  methods,  adds  2  n  ciphers  to  the  number, 
then  finds  the  square  root,  and  takes  thi§  as  a  numerator  of  a 
fraction  whose  denominator  is  1,  followed  by  n  ciphers.  The 
same  method  was  followed  by  Cardan,  but  it  failed  to  be  gen- 
erally adopted,  even  by  his  Italian  contemporaries ;  for  other- 
wise it  certainly  would  have  been  at  least  mentioned  by 
Cataldi  (died  1626)  in  a  work  devoted  exclusively  to  the  ex- 
traction of  roots.  Cataldi  finds  the  square  root  by  means  of 
continued  fractions  —  a  method  ingenious  and  novel,  but  for 
practical  purposes  inferior  to  Cardan's.  Orontius  Finseus,  in 
France,  and  William  Buckley  (died  about  1550),  in  England, 

1  For  an  example  of  square  root  by  the  scratch  method,  see  PEACOCK, 
p.  436. 

2  Consult  CANTOR,  L,  808;  PEACOCK,  p.  436. 


.      MODERN    TIMES  151 

extracted  the  square  root  in  the  same  way  as  Cardan.     In 

finding  the  square  root  of  10,  Finseus  adds  six  ciphers,  and 

lOOOOOOOC  3  1 162     concludes  ^as  shown  here.  The  3  1 162  expresses 

I  go      the  square  root  in  decimals.     A  new  branch 

9  |  720     of     arithmetic  —  decimal     fractions  —  thus 

I   60     stared  him  in  the  face,  as  it  had   many  of 

43  I  200     his  predecessors  and  contemporaries  !    But  he 

I     fiO 

sees  it  not;  he  is  thinking  of  sexagesimal 
fractions,  and  hastens  to  reduce  the  frac- 
tional part  to  sexagesimal  divisions  of  an  integer,1  thus, 
3  -  9'-  43".  12'".  What  was  needed  for  the  discovery  of  deci- 
mals in  a  case  like  this  ?  Observation,  keen  observation. 
And  yet  certain  philosophers  would  make  us  believe  that 
observation  is  not  needed  or  developed  in  mathematical 
study ! 

Close  approaches  to  the  discovery  of  decimals  were  made  in 
other  ways.  The  German  Christoff  Rudolff  performed  divis- 
ions by  10,  100,  1000,  etc.,  by  cutting  off  by  a  comma  ("mit 
einer  virgel ") 2  as  many  digits  as  there  are  zeros  in  the 
divisor. 

The  honour  of  the  invention  of  decimal  fractions  belongs  to 
Simon  Stevin  of  Bruges  in  Belgium  (1548-1620),  a  man 
remarkable  for  his  varied  attainments  in  science,  for  his  inde- 
pendence of  thought,  and  extreme  lack  of  respect  for  authority. 
It  would  be  interesting  to  know  exactly  how  he  came  upon  his 
great  discovery.  In  1584  he  published  in  Flemish  (later  in 
French)  an  interest  table.  "  I  hold  it  now  next  to  certain," 
says  De  Morgan,3  "that  the  same  convenience  which  has 
always  dictated  the  decimal  form  for  tables  of  compound 
interest  was  the  origin  of  decimal  fractions  themselves."  In 
1585  Stevin  published  his  La  Disme  (the  fourth  part  of  a 

i  PEACOCK,  p.  437.  2  CANTOR,  II.,  399. 

8  Arithmetical  Books,  p.  27. 


152  A   HISTORY   OF   MATHEMATICS 

French  work  on  mathematics),  covering  only  seven  pages,  in 
which  decimal  fractions  are  explained.  He  recognized  the 
full  importance  of  decimal  fractions,  and  applied  them  to  all 
the  operations  of  ordinary  arithmetic.  No  invention  is  per- 
fect at  its  birth.  Stevin' s  decimal  fractions  lacked  a  suitable 
notation.  In  place  of  our  decimal  point,  he  used  a  cipher;  to 
each  place  in  the  fraction  was  attached  the  corresponding 
index.  Thus,  in  his  notation,  the  number  5.912  would  be 

0123 

5912  or  5©9©1@2®.  These  indices,  though  cumbrous,  are 
interesting  because  herein  we  shall  find  the  principle  of 
another  important  innovation  made  by  Stevin  —  an  exponen- 
tial notation.  As  an  illustration  of  Stevin's  notation,  we 
append  the  following  division.1 

CO  (i)  (2)  (3)  (4)  (5)        (i)  (2)  He  was  enthusiastic,  not  only  over 

344362  by   96         decimal  fractions,  but  also  over  the 

I  decimal    division    of   weights   and 

*    $    $  measures.     He    considered    it    the 

y    „    «    7  duty   of  governments   to  establish 

P  (0)  (1)  (2)  (3)        , ,  ,     ,  .  ,        .          ,  , 

£    4"    4"    3    $    2(3587     the    latter.      As    to    decimals,    he 

9    fi    $    fi  says,  that,  while  their  introduction 

may   be    delayed,     "it    is    certain 

that  if  the  nature  of  man  in  the  future  remains  the  same  as 
it  is  now,  then  he  will  not  always  neglect  so  great  an  advan- 
tage." His  decimals  met  with  ready,  though  not  immediate, 
recognition.  His  La  Disme  was  translated  into  English  in 
1608  by  Richard  Norton.  A  decimal  arithmetic  was  published 
in  London,  1619,  by  Henry  Lyte.2  As  to  weights  and  measures, 
little  did  Stevin  suspect  that  two  hundred  years  would  elapse 
before  the  origin  of  the  metric  system ;  and  that  at  the  close 
of  the  nineteenth  century  England  and  the  New  World  would 
still  be  hopelessly  bound  by  the  chains  of  custom  to  the  use 

1  PEACOCK,  p.  440.  2  PEACOCK,  p.  440. 


MODERN   TIMES  158 

of  yards,  rods,  and  avoirdupois  weights.  But  we  still  hope  that 
the  words  of  John  Kersey  may  not  be  proved  prophetic :  "  It 
being  improbable  that  such  a  Reformation  will  ever  be  brought 
to  pass,  I  shall  proceed  in  directing  a  Course  to  the  Studious 
for  obtaining  the  frugal  Use  of  such  decimal  fractions  as  are 
in  his  Powers." 1 

After  Stevin,  decimals  were  used  on  the  continent  by  Joost 
Burgi,  a  Swiss  by  birth,  who  prepared  a  manuscript  on  arith- 
metic soon  after  1592,  and  by  Johann  Hartmann  Beyer,  who 
assumes  the  invention  as  his  own.  In  1603  he  published  at 
Frankfurt  on  the  Main  a  Logistica  Decimalis.  With  Burgi,  a 
zero  placed  underneath  the  digit  in  unit's  place  answers  as  a 
sign  of  separation.  Beyer's  notation  resembles  Stevin's,  but 
it  may  have  been  suggested  to  him  by  the  sexagesimal  notation 
then  prevalent.  He  writes  123.459872  thus : 

o     i     ii  in  iv    v    vi 

123  •  4  •  5  -  9  -  8  -  7  •  2. 

VI 

Again  he  writes  .000054  thus,  54,  and  remarks  that  these 
differ  from  other  fractions  in  having  the  denominator  written 
above  the  numerator.  The  decimal  point,  says  Peacock,  is 
due  to  Napier,  who  in  1617  published  his  Rabdologia  con- 
taining a  treatise  on  decimals,  wherein  die  decimal  point  is 
used  in  one  or  two  instances.  In  the  English  translation  of 
Napier's  Descriptio,  executed  by  Edward  Wright  in  1616,  and 
corrected  by  the  author,  the  decimal  point  occurs  on  the  first 
page  of  logarithmic  tables.  There  is  no  mention  of  decimals 
in  English  arithmetics  between  1619  and  1631.  Oughtred,  in 
1631,  designates  .56  thus,  0|56.  Albert  Girard,  a  pupil  of 
Stevin,  in  1629  uses  the  point  on  one  occasion.  John  Wallis, 
in  1657,  writes  12 1345,  but  afterwards  in  his  algebra  adopts  the 

1  KERSEY'S  WINGATE,  16th  Ed.,  London,  1735,  p.  119.  WILDERMCTH 
quotes  the  same  passage  from  the  2d  Ed.,  1668. 


154  A   HISTORY   OF   MATHEMATICS 

usual  point.  Georg  Andreas  Bockler,  in  his  Arithmetica  nova, 
Niirnberg,  1661,  uses  the  comma  in  place  of  the  point  (as  do 
the  Germans  at  the  present  time),  but  applies  decimals  only  to 
the  measurement  of  lengths,  surfaces,  and  solids.1  De  Morgan 2 
says  that  "it  was  long  before  the  simple  decimal  point  was 
fully  recognized  in  all  its  uses,  in  England  at  least,  and  on 
the  continent  the  writers  were  rather  behind  ours  in  this 
matter.  As  long  as  Oughtred  was  widely  used,  that  is,  till 
the  end  of  the  seventeenth  century,  there  must  have  been  a 
large  school  of  those  who  were  trained  to  the  notation  123 1456. 
To  the  first  quarter  of  the  eighteenth  century,  then,  we  must 
refer,  not  only  the  complete  and  final  victory  of  the  decimal 
point,  but  also  that  of  the  now  universal  method  of  performing 
the  operation  of  division  and  extraction  of  the  square  root." 
The  progress  of  the  decimal  notation,  and  of  all  other, 
is  interesting  and  instructive.  "  The  history  of  language  .  .  . 
is  of  the  highest  order  of  interest,  as  well  as  utility ;  its  sug- 
gestions are  the  best  lesson  for  the  future  which  a  reflecting 
mind  can  have."  (De  Morgan.) 

To  many  readers  it  will  doubtless  seem  that  after  the 
Hindu  notation  was  brought  to  perfection  probably  in  the 
ninth  century,  decimal  fractions  should  have  arisen  at  once 
in  the  minds  of  mathematicians,  as  an  obvious  extension  of  it. 
But 3  "  it  is  curious  to  think  how  much  science  had  attempted 
in  physical  research,  and  how  deeply  numbers  had  been  pon- 
dered before  it  was  perceived  that  the  all-powerful  simplicity 
of  the  '  Arabic  Notation '  was  as  valuable  and  as  manageable 
in  an  infinitely  descending  as  in  an  infinitely  ascending  pro- 
gression." 

The  experienced  teacher  has  again  and  again  made  observa- 

1  WILDERMUTH.  2  Arithmetical  Books,  p.  26. 

8  NAPIER,  MARK.  Memoirs  of  John  Napier  of  Merchiston,  Edinburgh, 
1834,  Chap.  IL 


MODERN   TIMES  155 

tions  similar  to  this,  on  the  development  of  thought,  in  watch- 
ing the  progress  of  his  pupils.  The  mind  of  the  pupil,  like 
the  mind  of  the  investigator,  is  bent  upon  the  attainment  of 
some  fixed  end  (the  solution  of  a  problem),  and  any  con- 
sideration not  directly  involved  in  this  immediate  aim 
frequently  escapes  his  vision.  Persons  looking  for  some 
particular  flower  often  fail  to  see  other  flowers,  no  matter 
how  pretty.  One  of  the  objects  of  a  successful  mathematical 
teacher,  as  of  a  successful  teacher  in  natural  science,  should 
be  to  habituate  students  to  keep  a  sharp  lookout  for  other 
things,  besides  those  primarily  sought,  and  to  make  these,  too, 
subjects  of  contemplation.  Such  a  course  develops  investi- 
gators, original  workers. 

The  miraculous  powers  of  modern  calculation  are  due  to 
three  inventions :  the  Hindu  Notation,  Decimal  Fractions,  and 
Logarithms.  The  invention  of  logarithms,  in  the  first  quarter  of 
the  seventeenth  century,  was  admirably  timed,  for  Kepler  was 
then  examining  planetary  orbits,  and  Galileo  had  just  turned 
the  telescope  to  the  stars.  During  the  latter  part  of  the  fif- 
teenth and  during  the  sixteenth  century,  German  mathema- 
ticians had  constructed  trigonometrical  tables  of  great  accuracy, 
but  this  greater  precision  enormously  increased  the  work  of  the 
calculator.  It  is  no  exaggeration  to  say  with  Laplace  that  the 
invention  of  logarithms  "by  shortening  the  labours  doubled 
the  life  of  the  astronomer."  Logarithms  were  invented  by 
John  Napier,  Baron  of  Merchiston,  in  Scotland  (1550-1617). 
At  the  age  of  thirteen,  Napier  entered  St.  Salvator  College, 
St.  Andrews.  An  uncle  once  wrote  to  Napier's  father,  "I 
pray  you,  Sir,  to  send  John  to  the  schools  either  of  France 
or  Flanders,  for  he  can  learn  no  good  at  home."  So  he  was 
sent  abroad.  In  1574  a  beautiful  castle  was  completed  for 
him  on  the  banks  of  the  Endrick.  On  the  opposite  side  of 
the  river  was  a  lint  mill,  and  its  clack  greatly  disturbed 


156  A   HISTORY   OF   MATHEMATICS 

Napier.  He  sometimes  desired  the  miller  to  stop  the  mill 
so  that  the  train  of  his  ideas  might  not  be  interrupted.1  In 
1608,  at  the  death  of  his  father,  he  took  possession  of  Mer- 
chiston  castle. 

Napier  was  an  ardent  student  of  theology  and  astrology,2 
and  delighted  to  show  that  the  pope  was  Antichrist.  More 
worthy  of  his  genius  were  his  mathematical  studies,  which  he 
pursued  as  pastime  for  over  forty  years.  Some  of  his  mathe- 
matical fragments  were  published  for  the  first  time  in  1839. 
The  great  object  of  his  mathematical  studies  was  the  simpli- 
fying and  systematizing  of  arithmetic,  algebra,  and  trigo- 
nometry. Students  in  trigonometry  remember  "  Napier's 
analogies,"  and  "  Napier's  rule  of  circular  parts,"  for  the  solu- 
tion of  spherical  right  triangles.  This  is,  perhaps,  "the 
happiest  example  of  artificial  memory  that  is  known."  In 
1617  was  published  his  Rabdologia,  containing  "Napier's 
rods"  or  "bones"3  and  other  devices  designed  to  simplify 
multiplication  and  division.  This  work  was  well  known  on 
the  continent,  and  for  a  time  attracted  even  more  attention 
than  his  logarithms.  As  late  as  1721  E.  Hatton,  in  his  arith- 

1  Diet.  Nat.  Biog. 

2  In  this  connection  the  title  of  the  following  book  is  interesting.     "  A 
Bloody  Almanack  Foretelling  many  certaine  predictions  which  shall  come 
to  passe  this  present  yeare  1647.     With  a  calculation  concerning  the 
time  of  the  day  of  Judgment,  drawne  out  and  published  by  that  famous 
astrologer,  the  Lord  Napier  of  Marcheston. "    For  this  and  for  a  catalogue 
of  Napier's  works,  see  MACDONALD'S  Ed.  of  NAPIER'S  Construction  of  the 
Wonderful  Canon  of  Logarithms,  1889. 

3  For  a  description  of  Napier's  bones,  see  article  "Napier,  John,"  in 
the  Encyclopedia  Britannica,  9th  Ed.     In  the  dedication  Napier  says, 
"  I  have  always  endeavoured  according  to  my  strength  and  the  measure 
of  my  ability  to  do  away  with  the  difficulty  and  tediousness  of  calcula- 
tions, the  irksomeness  of  which  is  wont  to  deter  very  many  from  the 
study  of  mathematics."    See  MACDONALD'S  Ed.  of  NAPIEK'S  Construction, 
p.  88. 


MODERN   TIMES  157 

metic,   takes   pains   to   explain    multiplication,    division,   and 
evolution  by  "Neper's  Bones  or  Rods." 

His  logarithms  were  the  result  of  prolonged,  unassisted 
and  isolated  speculation.  Nowadays  we  usually  say  that,  in 
n  =  bx,  x  is  the  logarithm  of  n  to  the  base  b.  But  in  the  time 
of  Napier  our  exponential  notation  was  not  yet  in  vogue. 
The  attempts  to  introduce  exponents,  made  by  Stifel  and 
Stevin,  were  not  yet  successful,  and  Harriot,  whose  algebra 
appeared  long  after  Napier's  death,  knew  nothing  of  indices. 
It  is  one  of  the  greatest  curiosities  of  the  history  of  science 
that  Napier  constructed  logarithms  before  exponents  were 
used.  That  logarithms  flow  naturally  from  the  exponential 
symbol  was  not  observed  until  much  later  by  Euler.1  What, 
then,  was  Napier's  line  of  thought  ? 

Let  AE  be  a  definite  line,  A'D'  a  line  extending  from  A' 
indefinitely.     Imagine  two  points  starting  at  the  same  moment  ; 
the  one  moving  from  A 
toward  Ej  the  other  from      i  -  1  -  1-_*  _  f 


A1  along  A'D'.     Let  the 

velocity  during  the   first     £      g;  -  £;  -  ft  - 

moment  be  the  same  for 

both.     Let  that  of  the  point  on  line   A'D'  be  uniform;   but 

the  velocity  of  the  point  on  AE  decreasing  in  such   a  way 

that  when  it  arrives  at  any  point  C,  its  velocity  is  proportional 

to  the  remaining  distance  CE.     If  the  first  point  moves  along 

a  distance  AC,  while  the  second  one  moves  over  a  distance 

A'C',  then  Napier  calls  A'C'  the  logarithm  of  CE. 

This  process  appears  strange  to  the  modern  student.  Let 
us  develop  the  theory  more  fully.  Assume  a  very  large  initial 
velocity  =  AE  =  (say)  v.  Then,  during  every  successive  short 

interval  or  moment  of  time,  measured  approximately  by   _,  the 

v 

1  J.  J.  WALKER,  "Influence  of  Applied  on  the  Progress  of  Pure 
Mathematics,"  Proceedings  Lond.  Math.  Soc.,  XXII.,  1890. 


158  A   HISTORY   OF   MATHEMATICS 

lower  point  will  travel  approximately  unit  distance,  which  is 

the  product  of  the  uniform  velocity  v  and  the  time  -  (nearly). 

v 

The  upper  point,  starting  likewise  with  a  velocity  v  =  AE, 
travels  during  the  first  moment  very  nearly  unit's  distance 

AB,  and  arrives  at  B  with  a  velocity  =  BE  =  v  —  1=1/1  —  -V 
During  the  second  moment  of  time  the  velocity  of  the  upper 

point  is  very  nearly  v  —  1,  hence  the  distance  BC  is  v~    ,  and 

v 

the  distance  CE  =  BE  —  BC  =  v  —  1  -  ^-—  ^  =  v(l  -  iV     The 

v  \       vj 

distance  of  the  point  from  E  at  the  end  of  the  third  moment 
is  similarly  found  to  be  v(  1  --  j  ,  and  after  the  vth  moment, 

f       IV 
vl  1  --  J  •     The  distances  from  E  of  the  upper  point  at  the 

end  of  successive  moments  are,  therefore,  represented  by  the 
first  of  the  two  following  series  (approximately), 


0,  1,  2,  3,         .-,          v. 

The  second  series  represents  at  the  end  of  corresponding 
intervals  of  time  very  nearly  the  distances  of  D'  from  A'. 
According  to  Napier,  the  numbers  in  the  lower  series  are  ap- 
proximately the  logarithms  of  the  corresponding  numbers  in 
the  upper  series.  Now  observe  that  the  lower  series  is  an 
arithmetical  progression  and  the  upper  a  geometrical  progres- 
sion. It  is  here  that  Napier's  discovery  comes  in  touch  with 
the  work  of  previous  investigators,  like  Archimedes  and  Stifel  ; 
it  is  here  that  the  continuity  between  the  old  and  the  new 
exists. 

The  relation  between  numbers  and  their  logarithms,  which 


MODERN    TIMES  159 

is  indicated  by  the  above  series,  is  found,  of  course,  in  the  loga- 
rithms now  in  general  use.  The  numbers  in  the  geometric 
series,  1,  10,  100,  1000,  have  for  their  common  logarithms  (to 
the  base  10),  the  numbers  in  the  arithmetic  series,  0,  1,  2,  3. 
But  observe  one  very  remarkable  peculiarity  of  Napier's  log- 
arithms: they  increase  as  the  numbers  themselves  decrease 
and  numbers  exceeding  v  have  negative  logarithms.  More 
over,  zero  is  the  logarithm,  not  of  unity  (as  in  modern  log- 
arithms), but  of  v,  which  was  taken  by  Napier  equal  to  107. 
Napier  calculated  the  logarithms,  not  of  successive  integral 
numbers,  from  1  upwards,  but  of  sines.  His  aim  was  to  sim- 
plify trigonometric  computations.  The  line  AE  was  the  sine 
of  90°  (i.e.  of  the  radius)  and  was  taken  equal  to  107  units. 
BE,  CE,  DE,  were  sines  of  arcs,  and  A'B',  A'C',  A'D'  their 
respective  logarithms.  It  is  evident  from  what  has  been 
said  that  the  logarithms  of  Napier  are  not  the  same  as 
the  natural  logarithms  to  the  base  e  =  2.718  •••.  This  dif- 
ference must  be  emphasized,  because  it  is  not  uncommon 
for  text-books  on  algebra  to  state  that  the  natural  logarithms 
were  invented  by  Napier.1  The  relation  existing  between 
natural  logarithms  and  those  of  Napier  is  expressed  by  the 
formula,2 

107 
Nap.  log  y  =  107  nat.  log  — 

It  must  be  mentioned  that  Napier  did  not  determine  the 

1  In  view  of  the  fact  that  German  writers  of  the  close  of  the  last  cen- 
tury were  the  first  to  point  out  this  difference,  it  is  curious  to  find  in 
Brockhaus'  Konversations  Lexikon  (1894),  article  "  Logarithmus,"  the 
statement  that  Napier  invented  natural  logarithms.     For  references  to 
articles  by  early  writers  pointing  out  this  error,  consult  DR.  S.  GUNTHER, 
Vermischte  Untersuchungen,  Chap.  V.,  or  my  Teaching  and  History  oj 
Mathematics  in  the  United  States,  p.  390. 

2  For  its  derivation  see  CA  JORI,  History  of  Mathematics,  p.  163. 


160  A    HISTORY  OF  MATHEMATICS 

base  to  Ms  system  of  logarithms.  The  notion  of  a  "base" 
not  only  never  suggested  itself  to  him,  but  in  fact  is  inapplica- 
ble to  his  system,  unless  it  is  modified  somewhat.1 

Napier's  great  invention  was  given  to  the  world  in  1614,  in 
a  work  entitled,  Mirifici  logarithmorum  canonis  description  In 
it  he  explained  the  nature  of  logarithms,  and  gave  a  logarith- 
mic table  of  the  natural  sines  of  a  quadrant  from  minute  to 
minute.  In  1619  appeared  Napier's  Mirifici  logarithmorum 
canonis  constructio,  as  a  posthumous  work,3  in  which  his 
method  of  calculating  logarithms  is  explained.4  The  follow- 


1  That  the  notion  of  a  "  base  "  may  become  applicable,  it  is  necessary 
that  zero  be  the  logarithm  of  1  and  not  of  107.  This  relation  log  1  =  0 
is  obtained  approximately  if  each  term  of  Napier's  geometric  series  and 
each  term  of  his  arithmetic  series  is  divided  by  v  or  107.  This  gives  us 


o  _ 

' 


Here  1  appears  as  the  logarithm  of  [  1  --  -  J     ,  which  is  nearly  equal 

to  e"1,  where  e  =  2.718  •••.  Hence  the  base  of  Napier's  logarithms,  as 
here  modified,  is  the  reciprocal  of  the  base  in  the  natural  system. 

2  From  a  note  at  the  end  of  the  table  of  logarithms  :  "  Since  the  calcu- 
lation of  this  table,  which  ought  to  have  been  accomplished  by  the  labour 
and  assistance  of  many  computors,  has  been  completed  by  the  strength 
and  industry  of  one  alone,  it  will  not  be  surprising  if  many  errors  have 
crept  into  it."  The  table  is  remarkably  accurate,  as  fewer  errors  have 
been  found  than  might  be  expected.  See  NAPIER'S  Construction  (MAC- 
DONALD'S  Ed.),  pp.  87,  90-96. 

8  It  has  been  republished  in  Latin  at  Paris,  1895.  An  English  transla- 
tion of  the  Construct™,  by  W.  R.  Macdonald,  appeared  in  Edinburgh, 
1889. 

4  For  a  brief  explanation  of  Napier's  mode  of  computation  see  CAN- 
TOR, II.,  p.  669. 


MODERN   TIMES 


161 


ing  is  a  copy  of  part  of   the  first  page  of  the  Descriptio  of 
1614: 


Gr.         0                                 +      - 

0 

Min. 

Sinus. 

Logarithm!. 

Differentiae. 

Logan  thmi. 

Sinus. 

0 

0 

Infinitum 

Infinitum 

0 

10000000 

60 

1 

2909 

81425681 

81425680 

1 

10000000 

59 

2 

5818 

74494213 

74494211 

2 

9999998 

58 

3 

8727 

70439564 

70439560 

4 

9999996 

57 

4 

11636 

67562746 

67562739 

7 

9999993 

56 

5 

14544 

65331315 

65331304 

11 

9999989 

55 

At  the  bottom  of  the  first  page  in  the  Descriptio,  on  the 
right,  is  the  figure  "89,"  for  89°.  In  the  columns  marked 
"  sinus,"  we  have  here  copied  the  natural  sines  of  0°,  0  to  5 
minutes,  and  of  89°,  55  to  60  minutes.  In  the  columns  marked 
"  logarithm i,"  are  the  logarithms  of  these  sines,  and  in  the 
column  "  differentiae  "  the  differences  between  the  logarithmic 
figures  in  the  two  columns.  Since  sin  x  =  cos  (90  —  x),  this 
semi-quadrantal  arrangement  of  the  tables  really  gives  all  the 
cosines  of  angles  and  their  logarithms.  Thus,  log  cos  0°  5' 
=  11  and  log  cos  89°  55'  ==  65331315.  Moreover,  since  log  tan  x 
=  —  log  cot  x  =  log  sin  x  —  log  cos  x,  the  column  marked  "  differ- 
entiae" gives  the  logarithmic  tangents,  if  taken  -+•>  and  the 
logarithmic  cotangents  if  taken  — . 

Napier's  logarithms  met  with  immediate  appreciation  both 
in  England  and  on  the  continent.  Henry  Briggs  (1556 1-1630), 
who  in  Napier's  time  was  professor  of  geometry  in  Gresham 
College,  London,  and  afterwards  professor  at  Oxford,  was 
struck  with  admiration  for  the  book.  "  Neper,  lord  of  Mark- 


1  The  Diet,  of  National  Biography  gives  1561  as  the  date  of  his  birth. 


162  A  HISTORY  OF  MATHEMATICS 

inston,  hath  set  my  head  and  hands  at  work  with  his  new  and 
admirable  logarithms.  I  hope  to  see  him  this  summer,  if  it 
please  God,  for  I  never  saw  a  book  which  pleased  me  better 
and  made  me  more  wonder/'  Briggs  was  an  able  mathe- 
matician, and  was  one  of  the  few  men  of  that  time  who  did 
not  believe  in  astrology.  While  Napier  was  a  great  lover  of 
this  pseudo-science,  "  Briggs  was  the  most  satirical  man  against 
it  that  hath  been  known,"  calling  it  "  a  system  of  groundless 
conceits."  Briggs  left  his  studies  in  London  to  do  homage 
to  the  Scottish  philosopher.  The  scene  at  their  meeting  is 
interesting.  Briggs  was  delayed  in  his  journey,  and  Napier 
complained  to  a  common  friend,  "Ah,  John,  Mr.  Briggs  will 
not  oome."  At  that  very  moment  knocks  were  heard  at  the 
gate,  and  Briggs  was  brought  into  the  lord's  chamber.  Almost 
one-quarter  of  an  hour  went  by,  each  beholding  the  other 
without  speaking  a  word.  At  last  Briggs  began :  "  My  lord, 
I  have  undertaken  this  long  journey  purposely  to  see  your 
person,  and  to  know  by  what  engine  of  wit  or  ingenuity  you 
came  first  to  think  of  this  most  excellent  help  in  astronomy, 
viz.  the  logarithms ;  but,  my  lord,  being  by  you  found  out,  I 
wonder  nobody  found  it  out  before,  when  now  known  it  is 
so  easy." *  Briggs  suggested  to  Napier  the  advantage  that 
would  result  from  retaining  zero  for  the  logarithm  of  the 
whole  sine,  but  choosing  107  for  the  logarithm  of  the  tenth 
part  of  that  same  sine,  i.e.  of  5°  44 '22".  Napier  said  that  he 
had  already  thought  of  the  change,  and  he  pointed  out  a 
slight  improvement  on  Briggs's  idea ;  viz.  that  zero  should  be 
the  logarithm  of  1,  and  107  that  of  the  whole  sine,  thereby 
making  the  characteristic  of  numbers  greater  than  unity 
positive  and  not  negative,  as  suggested  by  Briggs.  Briggs 
admitted  this  to  be  more  convenient.  The  invention  of  "  Brig- 

1  MARK  NAPIER'S  Memoirs  of  John  Napier,  1834,  p.  409. 


MODERN  TIMES  163 

gian  logarithms"  occurred,  therefore,  to  Briggs  and  Napier 
independently.  The  great  practical  advantage  of  the  new 
system  was  that  its  fundamental  progression  was  accommo- 
dated to  the  base,  10,  of  our  numerical  scale.  Briggs  devoted 
all  his  energies  to  the  construction  of  tables  upon  the  new 
plan.  Napier  died  in  1617?  with  the  satisfaction  of  having 
found  in  Briggs  an  able  friend  to  bring  to  completion  his 
unfinished  plans.  In  1624  Briggs  published  his  Arithmetica 
logarithmica,  containing  the  logarithms  to  14  places  of  num- 
bers, from  1  to  20,000  and  from  90,000  to  100,000.  The  gap 
from  20,000  to  90,000  was  filled  by  that  illustrious  suc- 
cessor of  Napier  and  Briggs,  Adrian  Vlacq,  of  Gouda  in 
Holland.  He  published  in  1628  a  table  of  logarithms  from 
1  to  100,000,  of  which  70,000  were  calculated  by  himself. 
The  first  publication  of  Briggian  logarithms  of  trigonometric 
functions  was  made  in  1620  by  Edmund  Gunter,  a  colleague 
of  Briggs,  who  found  the  logarithmic  sines  and  tangents  for 
every  minute  to  seven  places.  Gunter  was  the  inventor  of 
the  words  cosine  and  cotangent.  Briggs  devoted  the  last  years 
of  his  life  to  calculating  more  extensive  Briggian  logarithms 
of  trigonometric  functions,  but  he  died  in  1630,  leaving  his 
work  unfinished.  It  was  carried  on  by  the  English  Henry 
Gellibrand,  and  then  published  by  Vlacq  at  his  own  expense. 
Briggs  divided  a  degree  into  100  parts,  but  owing  to  the 
publication  by  Vlacq  of  trigonometrical  tables  constructed 
on  the  old  sexagesimal  division,  the  innovation  of  Briggs 
remained  unrecognized.  Briggs  and  Vlacq  published  four 
fundamental  works,  the  results  of  which  "have  never  been 
superseded  by  any  subsequent  calculations."  * 

1  For  further  information  regarding  logarithmic  tables,  consult  the 
articles  "Tables  (mathematical)"  in  the  Encyclopaedia  Britannica,  9th 
Ed.,  in  the  English  Cyclopaedia,  in  the  Penny  Cyclopaedia,  and  J.  W. 
L.  GLAISHER  in  the  report  of  the  committee  on  mathematical  tables, 


164  A   HISTORY   OF   MATHEMATICS 

We  have  pointed  out  that  the  logarithms  published  by 
Napier  are  not  the  same  as  our  natural  logarithms.  The  first 
appearance  of  natural  logarithms  (with  the  decimal  point 
omitted)  is  in  an  anonymous  Appendix  printed  in  the  1618 
edition  of  Edward  Wright's  translation  of  Napier's  Descriptio. 
This  interesting  Appendix  explains  methods  of  interpolation 
and  is  very  probably  written  by  William  Oughtred.  These 
interpolations  were  effected  with  the  aid  of  a  small  table  con- 
taining the  logarithms  of  72  sines.  In  it,  log  10  is  given  as 
230  2584.  In  modern  notation,  loge  10  =  2.302584. 

It  is  frequently  stated  that  natural  logarithms  appear  in  the 
New  Logarithmes,  published  at  London  by  a  London  teacher  of 
mathematics,  John  Speidell,  in  the  year  1619.  The  book  con- 
tains logarithms  of  sines,  tangents,  and  secants.  These  are  not 
quite  the  natural  logarithms,  on  account  of  complications  aris- 
ing from  the  fact  that  the  logarithms  appear  in  the  tables  as 
integral  numbers  and  also  that  the  natural  functions  (not 
printed  by  Speidell)  were  taken  as  integral  numbers.  Thus, 
Napier  gives  sin  30'  =  87265,  the  radius  being  107.  In  reality, 
sin  30'  =  .0087265.  The  natural  logarithm  of  this  fraction  is 
approximately  5.25861.  Adding  10  gives  5.25861.  Speidell 
writes  in  his  tables,  log  sin  30'  =  525  861.  The  relation  be- 
tween the  natural  logarithms  and  the  logarithms  in  Speidell's 
trigonometric  tables  is  shown  by  the  formula,  Sp.  log  x  = 

105flO  +  loge  — \  For  secants  and  the  latter  half  of  the  tan- 
gents the  addition  of  10  is  omitted.  In  Speidell's  table  log 
tan  89°  =  404  812,  the  natural  logarithm  of  tan  89°  being 
4.04812.1 

published  in  the  Report  of  the  British  Association  for  the  Advancement 
of  Science  for  1873,  pp.  1-175. 

1  On  SPEIDELL,  see  Quarterly  Journal  of  Pure  and  AppL  Math.,  Vol. 
46,  1915,  pp.  174-178  ;  article  "  Tables  "  in  the  English  Cyclopedia;  Re- 
port on  Tables  in  the  Report  of  the  British  Association  for  1873. 


MODERN   TIMES  165 

From  SpeidelFs  statements  it  is  evident  that  no  theoretical, 
but  purely  practical  reasons  induced  him  to  modify  Napier's 
tables.  He  desired  to  simplify  logarithms,  so  that  persons 
ignorant  of  the  algebraic  rules  of  addition  and  subtraction 
could  use  the  tables.  His  tables  of  1619  avoid  negative  num- 
bers, but  the  important  adaptation  to  the  Arabic  Notation, 
brought  about  in  the  Briggian  system,  escaped  him.  His  son, 
Euclid  Speidell,  says  that  he  "  at  last  concluded  that  the  deci- 
mal or  Briggs'  logarithms  were  the  best  sort  for  a  standard 
logarithm."  Editions  of  SpeidelFs  book  appear  to  have  been 
issued  in  1620,  1621,  1623,  1624,  1627,  1628,  but  not  all  by 
himself.  In  his  "  Briefe  Treatise  of  Sphaericall  Triangles  "  he 
mentions  and  complains  of  those  who  had  printed  his  work 
without  an  atom  of  alteration  and  yet  dispraised  it  in  their 
prefaces  for  want  of  alterations.  To  them  he  says  : 

"If  that  thou  canst  amend  it 
So  shall  the  arte  increase  : 
If  thou  canst  not :  commend  it, 
Else,  preethee  hould  thy  peace." 

This  unfair  treatment  of  himself  Speidell  attributes  to  his 
not  having  been  at  Oxford  or  Cambridge  —  "  not  hauing  scene 
one  of  the  Vniuersities." 

In  the  edition  of  1622  of  his  New  Logarithmes,  Speidell  pub- 
lished also  logarithms  of  the  numbers  1-1000 ;  these  are  genu- 
inely natural  logarithms,  except  for  the  omission  of  the  deci- 
mal point.  We  have  mentioned  the  very  small  table  of  1618, 
which  contains  72  logarithms.  SpeidelFs  is  the  earliest  more 
extensive  printed  table  of  natural  numbers.  There  is  a  column 
in  the  table  which  shows  that  he  means  to  use  his  table  in  cal- 
culations by  feet,  inches,  and  quarters.  Thus  the  number  775 
has  16-1-3  opposite  to  it,  there  being  775  quarter  inches  in 
16  feet  1  inch,  3  quarters.  This  is  an  interesting  example  of 
the  efforts  made  from  time  to  time  to  partially  overcome  the 


166  A    HISTORY   OF  MATHEMATICS 

inconvenience  arising  from  the  use  of  different  scales  in  the 
system  of  measures  from  that  of  our  notation  of  numbers.  At 
the  bottom  of  each  page  Speidell  inserts  the  logarithms  of  100 
and  1000  for  use  in  decimal  fractions. 

The  most  elaborate  system  of  natural  logarithmic  tables  is 
Wolfram's,  which  practically  gives  natural  logarithms  of  all 
numbers  from  1  to  10,000  to  48  decimal  places.  They  were 
published  in  1778  in  J.  C.  Schulze's  Sammlung.1  Wolfram 
was  a  Dutch  lieutenant  of  artillery  and  his  table  represents 
six  years  of  very  toilsome  work.  The  most  complete  table  of 
natural  logarithms,  as  regards  range,  was  published  by  the 
German  lightning  calculator  Zacharias  Dase,  at  Vienna,  in 
1850.  A  table  is  found  also  in  E-ees's  Cyclopaedia  (1819), 
article,  "  Hyperbolic  Logarithms." 

In  view  of  the  fact  that  all  early  efforts  were  bent,  not 
toward  devising  logarithms  beautiful  and  simple  in  .theory,  but 
logarithms  as  useful  as  possible  in  computation,  it  is  curious 
to  see  that  the  earliest  systems  are  comparatively  weak  in 
practical  adaptation,  although  of  great  theoretical  interest. 
Napier  almost  hit  upon  natural  logarithms,  whose  modulus 
is  unity,  while  Speidell  luckily  found  the  nugget. 

The  only  possible  rival  of  John  Napier  in  the  invention  of 
logarithms  was  the  Swiss  Joost  Btirgi  or  Justus  Byrgius 
(1552-1632).  In  his  youth  a  watchmaker,  he  afterwards  was 
at  the  observatory  in  Kassel,  and  at  Prague  with  Kepler.  He 
was  a  mathematician  of  power,  but  could  not  bring  himself  to 
publish  his  researches.  Kepler  attributes  to  him  the  discovery 
of  decimal  fractions  and  of  logarithms.  Btirgi  published  a 
crude  table  of  logarithms  six  years  after  the  appearance  of 
Napier's  Descriptio,  but  it  seems  that  he  conceived  the  idea 
and  constructed  that  table  as  early,  if  not  earlier,  than  Napier 

1  Article  "  Tables  "  in  the  English  Cyclopaedia. 


MODERN   TIMES  167 

did  his.1  However,  he  neglected  to  have  the  results  published 
until  after  Napier's  logarithms  (largely  through  the  influence 
of  Kepler)  were  known  and  admired  throughout  Europe. 

The  methods  of  computing  logarithms  adopted  by  Napier, 
Briggs,  Kepler,  Vlacq,  are  arithmetical  and  deducible  from  the 
doctrine  of  ratios.  After  the  large  logarithmic  tables  were 
once  computed,  writers  like  Gregory  St.  Vincent,  Newton, 
Nicolaus  Mercator,  discovered  that  these  computations  can  be 
performed  much  more  easily  with  aid  of  infinite  series.  In 
the  study  of  quadratures,  Gregory  St.  Vincent  (1584-1667)  in 
1647  found  the  grand  property  of  the  equilateral  hyperbola 
which  connected  the  hyperbolic  space  between  the  asymptotes 
with  the  natural  logarithms,  and  led  to  these  logarithms  being 
called  "  hyperbolic."  By  this  property,  Nicolaus  Mercator  in 
1668  arrived  at  the  logarithmic  series,  and  showed  how  the 
construction  of  logarithmic  tables  could  be  reduced  by  series 
to  the  quadrature  of  hyperbolic  spaces.2 

English  Weights  and  Measures 

At  as  late  a  period  as  the  sixteenth  century,  the  condition  of 
commerce  in  England  was  very  low.  Owing  to  men's  ignorance 
of  trade  and  the  general  barbarism  of  the  times,  interest  on 
money  in  the  thirteenth  century  mounted  to  an  enormous  rate. 
Instances  occur  of  50  %  rates.3  But  in  course  of  the  next  two 
centuries  a  reaction  followed.  Not  only  were  extortions  of 
this  sort  prohibited  by  statute,  but,  in  the  reign  of  Henry  VII., 
severe  laws  were  made  against  taking  any  interest  whatever, 

1  For  a  description  of  it  see  GERHARDT'S  Gesch.  d.  Math,  in  Deutschland, 
1877,  p.  119  ;  see  also  p.  75. 

2 For  various  methods  of  computing  logarithms,  consult  art.  "Loga- 
rithms" in  the  Encyclopaedia  Britannica,  9th  Ed. 

8  HUME'S  History  of  England,  Chap.  XII. 


168  A   HISTORY   OF   MATHEMATICS 

all  interest  being  then  denominated  usury.  That  commerce 
did  not  nourish  under  these  new  conditions  is  not  strange. 
But  after  the  middle  of  the  sixteenth  century  we  find  that 
the  invidious  word  usury  came  to  be  confined  to  the  taking  of 
exorbitant  or  illegal  interest ;  10  °/0  was  permitted.  What 
little  trade  existed  up  to  this  time  was  carried  on  mainly  by 
merchants  of  the  Hansa  Towns  —  Easterlings  as  they  were 
called.1 

But  in  the  latter  half  of  the  fifteenth  century  the  use  of 
gunpowder  was  introduced,  the  art  of  printing  invented, 
America  discovered.  The  pulse  and  pace  of  the  world  began 
to  quicken.  Even  in  England  the  wheels  of  commerce  were 
gradually  set  in  motion.  In  the  sixteenth  century  English 
commerce  became  brisk.  Need  was  felt  of  some  preparation 
for  business  careers.  Arithmetic  and  book-keeping  were 
introduced  into  Great  Britain. 

Questions  pertaining  to  money,  weights  and  measures  must 
of  necessity  have  received  some  attention  even  in  a  semi-civ- 
ilized community.  In  attempting  to  sketch  their  history,  we 
find  not  only  in  weights  and  measures,  but  also  in  English 
coins,  traces  of  Koman  influence.  The  Saxons  improved 
Koman  coinage.  That  of  William  the  Conqueror  was  appar- 
ently on  the  plan  adopted  by  Charlemagne  in  the  eighth 
century  and  is  supposed  to  be  derived  from  the  Romans  as 
regards  the  division  of  the  pound  into  20  shillings  and  the 
shilling  into  12  pence.  The  same  proportions  were  preserved 
in  the  lira  of  Italy,  libra  of  Spain,  and  lime  of  France,  all 
now  obsolete.  The  pound  adopted  by  William  the  Conqueror 
was  the  Saxon  Moneyer's  or  Tower  pound  of  5400  grains.2 

1  HUME'S  England,  Chap.  XXXV. 

2  P.   KELLY'S    Universal    Cambist,   London,    1835,    Vol.    I.,    p.    29. 
Much  of  our  information  regarding  English  money,  weights,  and  meas- 
ures is  derived  from  this  work. 


MODERN   TIMES  169 

Notice  that  the  vegetable  kingdom  supplied  the  primary  unit  of 
mass,  the  "  grain."  A  pound  in  weight  and  a  pound  in  tale  (that 
is,  in  reckoning)  were  the  same.  An  amount  of  silver  weighing 
one  pound  was  taken  to  be  worth  one  pound  in  money.  This 
explains  the  double  meaning  of  the  word  "  pound,"  first  as  a 
weight-unit,  secondly  as  a  money-unit.  Later  the  Troy  pound 
(5760  grains)  came  to  be  used  for  weighing  precious  metals. 
A  Troy  pound  of  silver  would  therefore  contain  21^  of  the 
shillings  mentioned  above.  Between  the  thirteenth  century 
and  the  beginning  of  the  sixteenth  -the  shilling  was  reduced 
to  ^  its  ancient  weight.  A  change  of  this  sort  would  probably 
have  been  disastrous,  except  for  the  practice  which  appears  to 
have  been  observed,  of  paying  by  weight  and  not  by  tale.  If 
we  except  one  short  period,  we  may  say  that  the  new  shilling 
was  permitted  to  vary  but  little  in  its  fineness.  In  1665,  in 
the  reign  of  Charles  II.,  a  Troy  pound  of  silver  yielded  62 
shillings,  in  1816  it  yielded  66  shillings.1  A  financial  expe- 
dient sometimes  resorted  to  by  governments  was  the  deprecia- 
tion of  the  currency,  by  means  of  issues  of  coin  which  contained 
much  less  silver  or  gold  than  older  coins,  but  which  had  nomi- 
nally the  same  value.  This  was  tried  by  Henry  VIII.,  who 
issued  money  in  which  the  silver  was  reduced  in  amount  by  J, 
later  1,  and  finally  f .  In  the  reign  of  Edward  VI.,  the  amount 
of  silver  was  reduced  by  f ,  so  that  a  coin  contained  only  \  the 
old  amount  of  silver.  Seventeen  years  after  the  first  of  these 
issues  Queen  Elizabeth  called  in  the  base  coin  and  put  money 
of  the  old  character  in  circulation.  Henry  VIII. 's  experiment 
was  disastrous,  and  reduced  England  "  from  the  position  of  a 
first  rate  to  that  of  a  third  rate  power  in  Europe  for  more  than 
a  century." 2 

1  KELLY,  Vol.  I.,  p.  29. 

2  Article,  "Finance,"  in  Encyclopaedia  Britannica,  9th  Ed.     Consult 
also  FRANCIS  A.  WALKER,  Money,  Chaps.  X.  and  XL 


170  A    HISTORY   OF   MATHEMATICS 

Interesting  are  the  etymologies  of  some  of  the  words  used 
in  connection  with  English  money.  The  name  sterling  seems 
to  have  been  introduced  through  the  Hansa  merchants  in 
London.  "  In  the  time  of  ...  King  Eichard  I.,  monie  coined 
in  the  east  parts  of  Germanic  began  to  be  of  especiall  request 
in  England  for  the  puritie  thereof,  and  was  called  Easterling 
monie,  as  all  the  inhabitants  of  those  parts  were  called  Easter- 
lings,  and  shortly  after  some  of  that  countrie,  skillful  in  mint 
matters,  .  .  .  were  sent  for  into  this  realme  to  bring  the  coine 
to  perfection ;  which  since  that  time  was  called  of  them  sterling, 
for  Easterling"  (Camden).  In  the  early  coinages  the  silver 
penny  or  sterling  was  minted  with  a  deep  cross.  When  it  was 
broken  into  four  parts,  each  was  called  &  fourth-ing  or  farthing, 
the  ing  being  a  diminutive.  Larger  silver  pieces  of  four  pence 
were  first  coined  in  the  reign  of  Edward  III.  They  were 
called  greats  or  groats.  In  1663,  in  the  reign  of  Charles  II., 
new  gold  coins  were  issued,  44|-  pieces  to  one  Troy  pound  of 
the  metal.  These  were  called  guineas,  after  the  new  country 
on  the  west  coast  of  Africa  whence  the  gold  was  brought.1 
The  guinea  varied  in  its  current  price  from  20  shillings  up  to 
30,  until  the  year  1717,  when,  on  Sir  Isaac  Newton's  recom- 
mendation, it  was  fixed  at  21  shillings,  its  present  value.2 

The  history  of  measures  of  weight  brings  out  the  curious 
fact  that  among  the  Hindus  and  Egyptians,  as  well  as  Italians, 
English,  and  other  Europeans,  the  basis  for  the  unit  of  weight 
lay  usually  in  the  grain  of  barley.  This  was  also  a  favourite 
unit  of  length.  The  lowest  subdivision  of  the  pound,  or  of 

1  THOMAS  DILWORTH  in  his  Schoolmaster's  Assistant,  1784  (first  edition 
about  1743)  praises  English  gold  as  follows,  p.  89  :  "In  England,  Sums  of 
Mony  are  paid  in  the  best  Specie,  viz. ,  Guineas,  by  which  Means  1000  1 
or  more  may  be  put  into  a  small  Bag,  and  conveyed  away  in  the  Pocket ; 
but  in  Sweden  they  often  pay  Sums  of  Mony  in  Copper,  and  the  Mer- 
chant is  obliged  to  send  Wheelbarrows  instead  of  Bags  to  receive  it." 

2  KELLY,  Vol.  I.,  p.  30. 


MODERN  TIMES  171 

other  similar  units,  was  usually  denned  as  weighing  the  same 
as  a  certain  number  of  grains  of  barley.  That  no  great  degree 
of  accuracy  could  be  secured  and  maintained  on  such  a  basis 
is  evident.  The  fact  that  the  writings  of  Greek  physicians 
were  widely  studied,  led  to  the  general  adoption  in  Europe 
of  the  Greek  subdivisions  of  the  litra  or  pound.  The  pound 
contained  12  ounces;  the  lower  subdivisions  being  the  drachm 
or  dram  ("  a  handful "),  the  gramma  ("  a  small  weight "),  and 
the  grain.1  The  Romans  translated  gramma  into  scriptulum 
or  scrupulum,  which  word  has  come  down  to  us  as  "  scruple." 
The  word  gramme  has  been  adopted  into  the  metric  system. 
The  Greeks  had  a  second  pound  of  16  ounces,  called  mina. 
The  subdivision  of  pounds  both  duodecimally  and  according 
to  the  fourth  power  of  two  is  therefore  of  ancient  date. 
During  the  Middle  Ages  there  was  in  Europe  an  almost 
endless  variety  of  different  sizes  of  the  pound,  as  also  of  the 
foot,  but  the  words  "  pound  "  and  "  foot "  were  adopted  by  all 
languages ;  thus  pointing  to  a  common  origin  of  the  measures. 
The  various  pounds  were  usually  divided  into  16  ounces ; 
sometimes  into  12.  The  word  "  pound  "  comes  from  the  Latin 
pondus.  The  word  "ounce"  is  the  Latin  uncia,  meaning  "a 
twelfth  part."  The  words  "ounce"  and  "inch"  have  one 
common  derivation,  the  former  being  called  uncia  librae,  (libra, 
pound),  and  the  latter  uncia  pedis  (pes,  foot).2 

On  English  standards  of  weights  and  measures  there  existed 
(we  are  told  by  an  old  bishop)  good  laws  before  the  Conquest, 
but  the  laws  then,  as  subsequently,  were  not  well  observed. 
The  Saxon  Tower  pound  was  retained  by  William  the  Con- 
queror, and  served  at  first  both  as  monetary  and  as  weight 
unit.  There  have  been  repeated  alterations  in  the  size  of  the 
pounds  in  use  in  England.  Moreover,  several  different  kinds 

*  PEACOCK,  p.  444.  2  KELLY,  Vol.  I.,  p.  20. 


172  A    HISTORY   OF   MATHEMATICS 

were  in  use  for  different  purposes  at  one  and  the  same  time. 
The  first  serious  attention  to  this  subject  seems  to  have  been 
given  in  1266,  statute  51  Henry  III.,  when,  "by  the  consent  of 
the  whole  realm  of  England  ...  an  English  penny,  called  a 
sterling,  round  and  without  any  clipping,  shall  weigh  32  wheat 
corns  in  the  midst  of  the  ear,  and  20  pence  do  make  an  ounce, 
and  12  ounces  one  pound,  and  8  pounds  do  make  a  gallon  of 
wine,  and  8  gallons  of  wine  do  make  a  London  bushel,  which 
is  the  eighth  part  of  a  quarter.'7  According  to  this  statement, 
the  pound  consisted  of,  and  was  determined  by,  the  weight  of 
7680  grains  of  wheat.  Moreover,  silver  coin  was  apparently 
taken  by  weight,  and  not  by  tale.  No  doubt  this  was  a 
grievance,  when  the  standard  of  weight  was  so  ill-established. 
The  pound  defined  above  we  take  to  be  the  same  as  the  Tower 
pound.  In  1527,  statute  18  Henry  VIII.,  the  Tower  pound, 
which  had  been  used  mostly  for  the  precious  metals,  was 
abolished,  and  the  Troy  established  in  its  place.  The  earliest 
statute  mentioning  the  Troy  pound  is  one  of  1414,  2  Henry  V. ; 
how  much  earlier  it  was  used  is  a  debated  question.  The  old 
Tower  pound  was  equivalent  to  ^f  of  the  Troy.  The  name 
Troy  is  generally  supposed  to  be  derived  from  Troyes  in 
France,  where  a  celebrated  fair  was  formerly  held,  and  the 
pound  was  used.  The  English  Committee  (of  1758)  on 
Weights  and  Measures  were  of  the  opinion  that  Troy  came 
from  the  monkish  name  given  to  London  of  Troy-novant, 
founded  on  the  legend  of  Brute.1  According  to  this,  Troy 

1  According  to  mythological  history,  Brute  was  a  descendant  of  JEneas 
of  ancient  Troy,  and  having  inadvertently  killed  his  father,  fled  to  Britain, 
founded  London,  and  called  it  Troy-novant  (New  Troy).  Spenser  writes, 
"Faery  Queen"  III.,  9, 

For  noble  Britons  sprong  from  Trojans  bold 

And  Troy-novant  was  built  of  old  Troyes  ashes  cold. 

See  BREWER'S  Die.  of  Phrase  and  Fable. 


MODERN    TIMES  173 

weight  means  "  London  weight."  At  about  the  same  period,  per- 
haps, the  avoirdupois  weight  was  established  for  heavy  goods. 
The  name  is  commonly  supposed  to  be  derived  from  the  French 
avoir-depots,  a  corrupt  spelling  introduced  for  avoir-de-pois, 
and  signifying  "goods  of  weight."1  In  the  earliest  statutes 
in  which  the  word  avoirdupois  is  used  (9  Edward  III.,  in  1335 ; 
and  27  Edward  III.,  in  1353),  it  is  applied  to  the  goods  them- 
selves, not  to  a  system  of  weights.  The  latter  statute  says, 
"  Forasmuch  as  we  have  heard  that  some  merchants  purchase 
avoirdupois  woollens  and  other  merchandise  by  one  weight 
and  sell  by  another,  .  .  .  we  therefore  will  and  establish  that 
one  weight,  one  measure,  and  one  yard  be  throughout  the  land, 
.  .  .  and  that  woollens  and  all  manner  of  avoirdupois  be 
weighed.  ..."  In  1532,  24  Henry  VIII.,  it  was  decreed  that 
"beef,  pork,  mutton,  and  veal  shall  be  sold  by  weight  called 
haverdupois." 2  Here  the  word  designates  weight.  According 
to  an  anonymous  arithmetic  of  1596,  entitled  "  The  Pathway  of 
Knowledge,"  the  "  pound  haberdepois  is  parted  into  16  ounces ; 
every  ounce  8  dragmes,  every  dragine  3  scruples,  every  scruple 
20  grains."  This  pound  contains  the  same  number  (7680)  of 
grains  as  the  statute  pound  of  1266  and  the  same  subdivisions 
of  the  ounce,  drachm,  and  scruple  as  our  present  apothecaries7 
weight.  The  number  of  grains  in  the  pennyweight  of  the  old 
pound  was  changed  from  32  to  24,  rendering  the  number  of 
grains  in  the  pound  5760.  Why  and  when  this  change  was 
made  we  do  not  know.  Cocker,  Wingate,  etc.,  say  in  their 
arithmetics  that  "  32  grains  of  wheat  make  24  artificial  grains."  3 
The  origin  of  our  apothecaries'  weight,  it  has  been  suggested,4 


1  MURRAY'S  English  Dictionary. 

2  JOHNSON'S  Universal  Cyclopaedia,  article  "Weights  and  Measures." 

3  COCKER'S  Arithmetic,  Dublin,  1714,  p.  13 ;  GEORGE  SHELLEY'S  Ed. 
of  WINGATE'S  Arithmetick,  16th  Ed.,  1735,  p.  7.      jf 

4  See  article  "  Weights  and  Measures  "  in  Penny  or  English  Cyclopaedia. 


174  A   HISTORY   OF   MATHEMATICS 

was  this :  Medicines  were  dispensed  by  the  old  subdivisions  of 
the  pound  (as  given  in  the  "  Pathway  of  Knowledge ")  and 
continued  to  be  so  after  the  standard  pound  (with  24  instead 
of  32  grains  in  the  pennyweight),  which  Queen  Elizabeth 
ordered  to  be  deposited  in  the  Exchequer  in  1588,  supplanted 
the  old  pound.  There  appear,  thus,  to  have  been  two  different 
pounds  avoirdupois,  an  old  and  a  new.  The  new  of  Queen 
Elizabeth  agreed  very  closely  with,  and  may  have  originated 
from,  an  old  merchant's  pound.1  It  must  be  remarked  that 
before  the  fifteenth  century,  and  even  later,  the  commercial 
weight  in  England  was  the  Amsterdam  weight,  which  was 
then  used  in  other  parts  of  Europe  and  in  the  East  and  West 
Indies.2  In  Scotland  it  has  been  in  partial  use  as  late  as  our 
century,  while  in  England  it  held  its  own  until  1815  in 
fixing  assize  of  bread.2  This  weight  tells  an  eloquent  story 
regarding  the  extent  of  the  Butch  trade  of  early  times. 

Other  kinds  of  pounds  are  mentioned  as  in  use  in  Great 
Britain.3  Their  variety  is  bewildering  and  confusing ;  their 
histories  and  relative  values  are  very  uncertain.  In  fact  a 
pound  of  the  same  denomination  often  had  different  values 
in  different  places.  The  "Pathway  of  Knowledge,"  1596, 
gives  five  kinds  of  pounds  as  in  use :  the  Tower,  the  Troy,  the 
"  haberdepoys,"  the  subtill,  and  the  f oyle.  The  subtill  pound 
was  used  by  assayers ;  the  f  oyle  was  ^  of  a  Troy  and  was  used 
for  gold  foil,  wire,  and  for  pearls.  In  case  of  gold  foil  and 
wire,  the  workman  probably  secured  his  profit  by  selling  £  of  a 
pound  at  the  price  of  a  pound  of  bullion.  Many  varieties  of 
measure  arose  from  the  practice  of  merchants,  who,  instead  of 

1  "  Weights  and  Measures"  in  Penny  or  in  English  Cyclopaedia. 

2  KELLY,  Vol.  II.,  p.  372,  note. 

8  Thus  DILWORTH,  in  his  Schoolmaster's  Assistant,  1784,  p.  38,  says : 
"Raw,  Long,  Short,  China,  Morea-Silk,  etc.,  are  weighed  by  a  great 
Pound  of  24  oz.  But  Ferret,  Filosella,  Sleeve-Silk,  etc.,  by  the  common 
Pound  of  16  oz." 


MODERN   TIMES  175 

varying  the  price  of  a  given  amount  of  an  article,  would  vary 
the  amount  of  a  given  name  (the  pound,  say)  at  a  given  price.1 

Ancient  measures  of  length  were  commonly  derived  from 
some  part  of  the  human  body.  This  is  seen  in  cubit  (length 
of  forearm),  foot,  digit,  palm,  span,  and  fathom.  Later  their 
lengths  were  defined  in  other  ways,  as  by  the  width  (or  length) 
of  barley  corns,  or  by  reference  to  some  arbitrarily  chosen 
standard,  carefully  preserved  by  the  government.  The  cubit 
is  of  much  greater  antiquity  than  the  foot.  It  was  used  by 
the  Egyptians,  Assyrians,  Babylonians,  and  Israelites.  It  was 
employed  in  the  construction  of  the  pyramid  of  Gizeh,  per- 
haps 3500  B.C.  The  foot  was  used  by  the  Greeks  and  the 
Romans.  The  Roman  foot  (=11.65  English  inches)  was  some- 
times divided  into  12  inches  or  uncice,  but  usually  into  4  palms 
(breadth  of  the  hand  across  the  middle  of  the  fingers)  and 
each  palm  into  4  finger-breadths.  Foot  rules  found  in  Roman 
ruins  usually  give  this  digital  division.2  By  the  Romans,  as 
also  by  the  ancient  Egyptians,  care  was  taken  to  preserve  the 
standards,  but  during  the  Middle  Ages  there  arose  great  diver- 
sity in  the  length  of  the  foot. 

As  previously  stated,  the  early  English,  like  the  Hindus  and 
Hebrews,  used  the  barley  corn  or  the  wheat  corn,  in  determin- 
ing the  standards  of  length  and  of  weight.  If  it  be  true  that 
Henry  I.  ordered  the  yard  to  be  of  the  length  of  his  arm,  then 
this  is  an  exception.  It  has  been  stated  that  the  Saxon  yard 
was  39.6  inches,  and  that,  in  the  year  1101,  it  was  shortened 
by  Henry  I.  to  the  length  of  his  arm.  The  earliest  English 
statute  pertaining  to  units  of  length  is  that  of  1324  (17  Edward 
II.),  when  it  was  ordained  that  "  three  barley  corns,  round  and 
dry,  make  an  inch,  12  inches  one  foot,  three  feet  a  yard." 
Here  the  lengths  of  the  barley  corns  are  taken,  while  later,  in 

1  "  Weights  and  Measures"  in  English  Cyclopaedia. 

2  DE  MORGAN,  Arith.  Books,  p.  5. 


176  A   HISTORY   OF    MATHEMATICS 

the  sixteenth  century,  European  writers  take  the  breadth;  thus, 
64  breadths  to  one  "  geometrical  "  foot  (Clavius).  The  breadth 
was  doubtless  more  definite  than  the  length,  especially  as  the 
word  "round"  in  the  law  of  1324  leaves  a  doubt  as  to  how 
much  of  the  point  of  the  grain  should  be  removed  before  it  can 
be  so  called. 

It  would  seem  as  though  uniformity  in  the  standards  should 
be  desired  by  all  honest  men,  and  yet  the  British  government 
has  always  experienced  great  difficulty  in  enforcing  it.  To  be 
sure,  the  provisions  of  law  often  increased  the  confusion. 
Thus,  in  1437,  by  statute  15  Henry  VI.,  the  alnager,  or  measurer 
by  the  ell,  is  directed  "  to  procure  for  his  own  use  a  cord  twelve 
yards  twelve  inches  long,  adding  a  quarter  of  an  inch  to  each 
quarter  of  a  yard."  This  law  marks  the  era  when  the  woollen 
manufacture  became  important,  and  the  law  was  intended  to 
make  certain  the  hitherto  vague  custom  of  allowing  the  width 
of  the  thumb  on  every  yard,  for  shrinkage.  In  1487,  as  if  to 
repeal  this,  it  was  ordained  that  "  cloths  shall  be  wetted  before 
they  are  measured,  and  not  again  stretched."  But  in  a  later 
year  the  older  statute  is  again  followed.1 

There  was  no  defined  relation  between  the  measures  of 
length  and  of  capacity,  until  1701.  The  statute  for  that  year 
declares  that  "the  Winchester  bushel  shall  be  round  with  a 
plane  bottom,  18 J  inches  wide  throughout,  and  8  inches  deep." 
In  measures  of  capacity  there  was  greater  diversity  than  in 
the  units  of  length  and  weight,  notwithstanding  the  fact  that 
in  the  laws  of  King  Edgar,  nearly  a  century  before  the  Con- 
quest, an  injunction  was  issued  that  one  measure,  the  Winches- 
ter, should  be  observed  throughout  the  realm.  Heavy  fines 
were  imposed  later  for  using  any  except  the  standard  bushel, 
but  without  avail.  The  history 2  of  the  wine  gallon  illustrates 

1  North  American  Review,  No.  XCVII.,  October,  1837. 

2  North  American  Review,  No.  XCVII. 


MODERN   TIMES  177 

how  standards  are  in  danger  of  deterioration.  By  a  statute  of 
Henry  III.  there  was  but  one  legal  gallon  —  the  wine  gallon. 
Yet  about  1680  it  was  discovered  that  for  a  long  time  importers 
of  wine  paid  duties  on  a  gallon  of  272  to  282  cu.  in.,  and  sold 
the  wine  by  one  of  capacity  varying  from  224  to  231  cu.  in. 

The  British  Committee  of  1758  on  weights  and  measures 
seemed  to  despair  of  success  in  securing  uniformity,  saying 
"that  the  repeated  endeavours  of  the  legislature,  ever  since 
Magna  Charta,  to  compel  one  weight  and  one  measure  through- 
out the  realm  never  having  proved  effectual,  there  seems  little 
to  be  expected  from  reviving  means  which  experience  has 
shown  to  be  inadequate." 

The  latter  half  of  the  eighteenth  and  the  first  part  of  the 
nineteenth  century  saw  the  wide  distribution  in  England  of 
standards  scientifically  defined  and  accurately  constructed. 
Nevertheless,  as  late  as  1871  it  was  stated  in  Parliament  that 
in  certain  parts  of  England  different  articles  of  merchandise 
were  still  sold  by  different  kinds  of  weights,  and  that  in 
Shropshire  there  were  actually  different  weights  employed 
for  the  same  merchandise  on  different  market-days.1 

The  early  history  of  our  weights  and  measures  discloses  the 
fact  that  standards  have  been  chosen,  as  a  rule,  by  the  people 
themselves,  and  that  governments  stepped  in  at  a  later  period 
and  ordained  certain  of  the  measures  already  in  use  to  be  legal, 
to  the  exclusion  of  all  others.  Measures  which  grow  directly 
from  the  practical  needs  of  the  people  engaged  in  certain  occu- 
pations have  usually  this  advantage,  that  they  are  of  conven- 
ient dimensions.  The  furlong  ("furrow-long")  is  about  the 
average  length  of  a  furrow;  the  gallon  and  hogshead  have 
dimensions  which  were  well  adapted  for  practical  use ;  shoe- 
makers found  the  barley  corn  a  not  unsuitable  subdivision  of 

1  JOHNSON'S  Universal  Cyclop  ,  Art.  "  Weights  and  Measures.1' 


178  A   HISTORY   OF   MATHEMATICS 

the  inch  in  measuring  the  length  of  a  foot.  A  very  remark- 
able example  of  the  convenient  selection  of  units  is  given  by 
De  Morgan : l  That  the  tasks  of  those  who  spin  might  be  cal- 
culated more  readily,  the  sack  of  wool  was  made  13  tods  of 
28  pounds  each,  or  364  pounds.  Thus,  a  pound  a  day  was  a 
tod  a  month,  and  a  sack  a  year.  But  where  are  the  Sundays 
and  holidays  ?  It  looks  as  though  the  weary  "  spinster  "  was 
obliged  to  put  in  extra  hours  on  other  days,  that  she  might 
secure  her  holiday.  Again,  "The  Boke  of  Measuryng  of 
Lande,"  by  Sir  Richarde  de  Benese  (about  1539),  suggests  to 
De  Morgan  the  following  passage : 1  "  The  acre  is  four  roods, 
each  rood  is  ten  daye-workes,  each  daye-worke  four  perches. 
So  the  acre  being  40  daye-workes  of  4  perches  each,  and  the 
mark  40  groats  of  4  pence  each,  the  aristocracy  of  money  and 
that  of  land  understood  each  other  easily."  In  a  system  like 
the  French,  systematically  built  up  according  to  the  decimal 
scale,  simple  relations  between  the  units  for  time,  for  amount 
of  work  done,  and  for.  earnings,  do  not  usually  exist.  This 
is  the  only  valid  objection  which  can  be  urged  against  a  sys- 
tem like  the  metric.  On  the  other  hand,  the  old  system  needs 
readjustment  of  its  units  every  time  that  some  invention  brings 
about  a  change  in  the  mode  of  working  or  a  saving  of  time, 
and  new  units  must  be  invented  whenever  a  new  trade  springs 
up.  Unless  this  is  done,  systems  on  the  old  plan  are,  if  not 
ten  thousand  times,  seven  thousand  six  hundred  forty-seven 
times  worse  than  the  metric  system.  Again,  the  old  mode  of 
selecting  units  leads  to  endless  varieties  of  units,  and  to  such 
atrocities  as  7.92  inches  =  1  link,  5^-  yards  =  1  rod,  16^  feet 
=  1  pole,  43560  sq.  feet  =  1  acre,  1^  hogshead  =  1  punch. 

The   advantages   secured    by  having   a  uniform   scale    for 
weights  and  measures,  which  coincides  with  that  of  the  Arabic 

1  Arithmetical  Books,  p.  18. 


MODERN   TIMES  179 

Notation,  are  admitted  by  all  who  have  given  the  subject  due 
consideration.  Among  early  English  writers  whose  names  are 
identified  with  attempted  reforms  of  this  sort  are  Edmund 
Gunter  and  Henry  Briggs,  who,  for  one  year,  were  colleagues 
in  Gresham  College,  London.  No  doubt  they  sometimes 
met  in  conference  on  this  subject.  Briggs  divided  the  degree 
into  100,  instead  of  60,  minutes ;  Gunter  divided  the  chain 
into  100  links,  and  chose  it  of  such  length  that  "  the  work  be 
more  easie  in  arithmetick ;  for  as  10  to  the  breadth  in  chains, 
so  the  length  in  chains  to  the  content  in  acres."  Thus,  y1^  the 
product  of  the  length  and  breadth  (each  expressed  in  chains) 
of  a  rectangle  gives  the  area  in  acres. 

The  crowning  achievement  of  all  attempts  at  reform  in 
weights  and  measures  is  the  metric  system.  It  had  its  origin 
at  a  time  when  the  French  had  risen  in  fearful  unanimity,  de- 
termined to  destroy  all  their  old  institutions,  and  upon  the 
ruins  of  these  plant  a  new  order  of  things.  Finally  adopted  in 
France  in  1799,  the  metric  system  has  during  the  present 
century  displaced  the  old  system  in  nearly  every  civilized 
realm,  except  the  English-speaking  countries.  So  easy  and 
superior  is  the  system,  that  no  serious  difficulty  has  been  en- 
countered in  its  introduction,  wherever  the  experiment  has 
been  tried.  The  most  pronounced  opposition  to  it  was  shown 
by  the  French  themselves.  The  ease  with  which  the  change 
has  been  made  in  other  countries,  in  more  recent  time,  is  due, 
in  great  measure,  to  the  fact  that,  before  its  adoption,  the 
metric  system  had  been  taught  in  many  of  the  schools. 

Rise  of  the  Commercial  School  of  Arithmeticians  in  England 

Owing  to  the  backward  condition  of  England,  arithmetic  was 
cultivated  but  little  there  before  the  sixteenth  century.  In 
the  fourteenth  century  the  Hindu  numerals  began  to  appear  in 


180  A    HISTORY    OF    MATHEMATICS 

Great  Britain.  A  single  instance  of  their  use  in  the  thir 
teenth  century  is  found  in  a  document  of  1282,  where  the 
word 1  trium  is  written  3  um.  In  1325  there  is  a  warrant  from 
Italian  merchants  to  pay  40  pounds ;  the .  body  of  the  docu- 
ment contains  Roman  numerals,  but  on  the  outside  is  endorsed 
by  one  of  the  Italians,  13X<7>  that  is,  1325.  The  5  appears  in- 
complete and  inverted,  resembling  the .  old  5  in  the  Bamberg 
Arithmetic  of  1483,  and  the  5  of  the  apices  of  Boethius.  The 
2  has  a  slight  resemblance  to  the  2  in  the  Bamberg  Arith- 
metic. The  Hindu  numerals  of  that  time  were  so  different 
from  those  now  in  use,  and  varied  so  greatly  in  their  form, 
that  persons  unacquainted  with  their  history  are  apt  to  make 
mistakes  in  identifying  the  digits.  A  singular  practice  of 
high  antiquity  was  the  use  of  the  old  letter  fj  for  5.  Curious 
errors  sometimes  occur  in  notation.  Thus,  X2  for  12 ;  XXXI, 
or  301,  for  31.  The  new  numerals  are  not  found  in  the  books 
printed  by  Caxton,  but  in  the  Myrrour  of  the  World,  issued  by 
him  in  1480,  there  is  a  wood-cut  of  an  arithmetician  sitting  before 
a  table  on  which  are  tablets  with  Hindu  numerals  upon  them. 

The  use  of  Hindu  numerals  in  England  in  the  fif teenth  »cen- 
tury  is  rather  exceptional.  Until  the  middle  of  the  sixteenth 
century  accounts  were  kept  by  most  merchants  in  Roman 
numerals.  The  new  symbols  did  not  find  widespread  accept- 
ance till  the  publication  of  English  arithmetics  began.  As  in 
Italy,  so  in  England,  the  numerals  were  used  by  mercantile 
houses  much  earlier  than  by  monasteries  and  colleges. 

The  first  important  arithmetical  work  of  English  authorship 
was  published  in  Latin  in  1522  by  Cuthbert  Tonstall  (1474- 
1559).  No  earlier  name  is  known  to  us  excepting  that  of 
John  Norfolk,  who  wrote,  about 2 1340,  an  inferior  treatise  on 

1  Our  account  of  the  numerals  in  England  is  taken  from  JAMES  A. 
PICTON'S  article  On  the  Origin  and  History  of  the  Numerals,  1874. 

2  W.  W.  R.  BALL,  History  of  the  Study  of  Mathematics  at  Cambridge^ 
Cambridge,  1889,  p.  7. 


MODERN    TIMES  181 

progressions  which  was  printed  in  1445,  and  reissued  by  Halli- 
well  in  his  Kara  Mathematica,  London,  1841.  Norfolk  con- 
founds arithmetical  and  geometrical  progressions,  and  confines 
himself  to  the  most  elementary  considerations. 

Tonstall  studied  at  Oxford,  Cambridge,  and  Padua,  and 
drew  freely  from  the  works  of  Pacioli  and  Regiomontanus. 
Reprints  of  his  arithmetic,  De  arte  supputandi,1  appeared  in 
England  and  France,  and  yet  it  seems  to  have  been  but  little 
known  to  succeeding  English  writers.2  The  author  states  that 
some  years  previous  he  had  dealings  in  money  (argentariis) 
and,  not  to  get  cheated,  had  to  study  arithmetic.  He  read 
everything  on  the  subject  in  every  language  that  he  knew,  and 
spent  much  time,  he  says,  in  licking  what  he  found  into  shape, 
ad  ursi  exemplum,  as  the  bear  does  her  cubs.  According  to 
De  Morgan2  this  book  is  "decidedly  the  most  classical  which 
ever  was  written  on  the  subject  in  Latin,  both  in  purity  of 
style  and  goodness  of  matter."  The  book  is  a  "  farewell  to 
the  sciences  "  on  the  author's  appointment  to  the  bishopric  of 
London.  A  modern  critic  would  say  that  there  is  not  enough 
demonstration  in  this  arithmetic,  but  Tonstall  is  a  very  Euclid 
by  the  side  of  most  of  his  contemporaries.  Arithmetical  results 
frequently  needed,  he  arranges  in  tables.  Thus,  he  gives  the 
multiplication  table  in  form  of  a  square,  also  addition,  subtrac- 
tion, and  division  tables,  and  the  cubes  of  the  first  10  numbers. 

For3  J  of  ^  of  ^  he  has  the  notation  J  I  \.  Interesting  is 
his  discussion  of  the  multiplication  of  fractions.  We  must 
here  premise  that  Pacioli  (like  many  a  school-boy  of  the 
present  day)  was  greatly  embarrassed 4  by  the  use  of  the  term 

1  In  this  book,  the  pages  are  not  numbered.    The  earliest  known  work 
in  which  the  Hindu  numerals  are  used  for  numbering  the  pages  is  one 
printed  in  1471  at  Cologne.     See  UNGER,  p.  16,  and  KASTNER,  Vol.  I., 
p.  94.  3  CANTOR,  II.,  476. 

2  DE  MORGAN,  Arithmetical  Books,  p.  13.  4  PEACOCK,  p.  439. 


182  A   HISTORY   OF   MATHEMATICS 

"  multiplication  "  in  case  of  fractions,  where  the  product  is  less 
than  the  multiplicand.  That  "  multiply  "  means  "  increase  " 
he  proves  from  the  Bible :  "  Be  fruitful  and  multiply,  and  re- 
plenish the  earth  "  (Gen.  i :  28)  ;  "  I  will  multiply  thy  seed  as 
the  stars  of  the  heaven"  (Gen.  xxii:  17).  But  how  is  this  to 
be  reconciled  with  the  product  of  fractions?  In  this  way: 
the  unit  in  the  product  is  of  greater  virtue  or  significance; 
thus,  if  \  and  \  are  the  sides  of  a  square,  then  \  represents 
the  area  of  the  square  itself.  Later  writers  encountered  the 
same  difficulty,  but  were  not  always  satisfied  with  Pacioli's 
explanation.  Tonstall  discusses  the  subject  with  unusual 
clearness.  He  takes  -J  x  f  =  T%.  "  If l  you  ask  the  reason  why 
this  happens  thus,  it  is  this,  that  if  the  numerators  alone  are 
multiplied  together  the  integers  appear  to  be  multiplied 
together,  and  thus  the  numerator  would  be  increased  too 
much.  Thus,  in  the  example  given,  when  2  is  multiplied  into 
3,  the  result  is  6,  which,  if  nothing  more  were  done,  would 
seem  to  be  a  whole  number ;  however,  since  it  is  not  the  inte- 
ger 2  that  must  be  multiplied  by  3,  but  f  of  the  integer  1  that 
must  be  multiplied  by  f  of  it,  the  denominators  of  the  parts 
are  in  like  manner  multiplied  together ;  so  that,  finally,  by  the 
division  which  takes  place  through  multiplication  of  the  de- 
nominators (for  by  so  much  as  the  denominator  increases,  by 
so  much  are  the  parts  diminished),  the  increase  of  the  numer 
ator  is  corrected  by  as  much  as  it  had  been  augmented  more 
than  was  right,  and  by  this  means  it  is  reduced  to  its  proper 
value." 

This  dispute  is  instanced  by  Peacock  as  a  curious  example 
of  the  embarrassment  arising  when  a  term,  restricted  in  mean- 
ing, is  applied  to  a  general  operation,  the  interpretation  of 
which  depends  upon  the  kinds  of  quantities  involved.  The 

1  We  are  translating  Tonstall's  Latin,  quoted  by  PEACOCK,  p.  439. 


MODERN   TIMES  183 

difficulty  which  is  encountered  in  multiplication  arises  also 
in  the  process  of  division  of  fractions,  for  the  quotient  is 
larger  than  the  dividend.  The  explanation  of  the  paradox 
calls  for  a  clear  insight  into  the  nature  of  a  fraction.  That, 
in  the  historical  development,  multiplication  and  division 
should  have  been  considered  primarily  in  connection  with 
integers,  is  very  natural.  The  same  course  must  be  adopted 
in  teaching  the  young.  First  come  the  easy  but  restricted 
meanings  of  multiplication  and  division,  applicable  to  whole 
numbers.  In  due  time  the  successful  teacher  causes  students 
to  see  the  necessity  of  modifying  and  broadening  the  meanings 
assigned  to  the  terms.  A  similar  plan  has  to  be  followed  in 
algebra  in  connection  with  exponents.  First  there  is  given  an 
easy  definition  applicable  only  to  positive  integral  exponents. 
Later,  new  meanings  must  be  sought  for  fractional  and  nega- 
tive exponents,  for  the  student  sees  at  once  the  absurdity,  if 
we  say  that  in  xz,  x  is  taken  one-half  times  as  a  factor. 
Similar  questions  repeatedly  arise  in  algebra. 

Of  course,  Tonstall  gives  the  English  weights  and  meas- 
ures ;  he  also  compares  English  money  with  the  French,  etc. 

It  is  worthy  of  remark  that  Tonstall,  to  prevent  Tyndale's 
translation  of  the  New  Testament  from  spreading  more  widely 
among  the  masses,  is  said  to  have  once  purchased  and  burned 
all  the  copies  which  remained  unsold.  But  the  bishop's 
money  enabled  Tyndale,  the  following  year,  to  bring  out  a 
second  and  more  correct  edition ! 

A  quarter  of  a  century  after  the  first  appearance  of  Ton- 
stall's  arithmetic,  were  issued  the  writings  of  Robert  Recorde 
(1510-1558).  Educated  at  Oxford  and  Cambridge,  he  ex- 
celled in  mathematics  and  medicine.  He  taught  arithmetic 
at  Oxford,  but  found  little  encouragement,  notwithstanding 
the  fact  that  he  was  an  exceptional  teacher  of  this  subject. 
Migrating  to  London,  he  became  physician  to  Edward  VI., 


184  A    HISTORY   OF    MATHEMATICS 

and  later  to  Queen  Mary.  It  does  not  seem  that  he  met  with 
recompense  at  all  adequate  to  his  merits,  for  afterwards  he  was 
confined  in  prison  for  debt,  and  there  died.  He  wrote  several 
works,  of  which  we  shall  notice  his  arithmetic  and  (elsewhere) 
his  algebra.  It  is  stated  that  Recorde  was  the  earliest  Eng- 
lishman who  accepted  and  advocated  the  Copernican  theory.1 

His  arithmetic,  The  Grounde  of  Artes,  was  published  in 
1540.  Unlike  Tonstall's,  it  is  written  in  English  and  contains 
the  symbols  +,  — ,  Z-  The  last  symbol  he  uses  to  denote 
equality.  In  his  algebra  it  is  modified  into  our  familiar 
sign  =.  These  three  symbols  are  not  used  except  toward  the 
last,  under  "  the  rule  of  Falsehode."  He  says 2  "  +  whyche 
betokeneth  too  muche,  as  this  line,  — ,  plaine  without  a  crosse 
line,  betokeneth  too  little."  The  work  is  written  in  the  form 
of  a  dialogue  between  master  and  pupil.  Once  the  pupil  says, 
"  And  I  to  youre  authoritie  my  witte  doe  subdue,  whatsoever 
you  say,  I  take  it  for  true,"  whereupon  the  master  replies 
that  this  is  too  much,  "  thoughe  I  mighte  of  my  Scholler  some 
credence  require,  yet  except  I  shew  reason,  I  do  it  not  desire." 
Notice  here  the  rhyme,  which  sometimes  occurs  in  the  book, 
though  the  verse  is  not  set  off  into  lines  by  the  printer.  In 
all  operations,  even  those  with  denominate  numbers,  he  tests 
the  results  by  "  casting  out  the  9's." 3  The  scholar  complains 
that  he  cannot  see  the  reason  for  the  process.  "No  more 
doe  you  of  manye  things  else,"  replies  the  master,  arguing 
that  one  must  first  learn  the  art  by  concisely  worded  rules, 
before  the  reason  can  be  grasped.  This  is  certainly  sound 

1  BALL'S  Mathematics  at  Cambridge,  p.  18. 

2  CANTOR,  II.,  p.  477-480. 

8  The  insufficiency  of  this  test  is  emphasized  by  Cocker  as  follows : 
"But  there  may  be  given  a  thousand  (nay  infinite)  false  products  in 
multiplication,  which  after  this  manner  may  be  proved  to  be  true,  and 
therefore  this  way  of  proving  doth  not  deserve  any  example,"  Arithme- 
tick,  28th  Ed.,  1714,  p.  50. 


MODERN    TIMES  185 

advice.  The  method  of  "  casting  out  the  9's  "  is  easily  learned, 
but  the  reason  for  it  is  beyond  the  power  of  the  young  student. 
It  is  not  contrary  to  sound  pedagogy  sometimes  to  teach 
merely  the  facts  or  rules,  and  to  postpone  the  reasoning  to 
a  later  period.  Whoever  teaches  the  method  and  the  reason- 
ing in  square  root  together  is  usually  less  successful  than  he 
who  teaches  the  two  apart,  one  immediately  after  the  other. 
It  is,  we  believe,  the  usual  experience,  both  of  individuals 
and  of  nations,  that  the  natural  order  of  things  is  facts  first, 
reasons  afterwards.  This  view  is  no  endorsement  of  mere 
memory-culture,  which  became  universal  in  England  after 
the  time  of  Tonstall  and  Recorde. 

Recorde  gives  the  Rule  of  Three  (or  "the  golden  rule"), 
progressions,  alligation,  fellowship,  and  false  position.  He 
thinks  it  necessary  to  repeat  all  the  rules  (like  the  rule  of 
three,  fellowship,  etc.)  for  fractions.  This  practice  was  preva- 
lent then,  and  for  the  succeeding  250  years.  He  was  particu- 
larly partial  to  the  rule  of  false  position  ("  rule  of  Falsehode  ") 
and  remarks  that  he  was  in  the  habit  of  astonishing  his  friends 
by  proposing  hard  questions  and  deducing  the  correct  answer 
by  taking  the  chance  replies  of  "suche  children  or  ydeotes 
as  happened  to  be  in  the  place." 

It  is  curious  to  find  in  Recorde  a  treatise  on  reckoning  by 
counters,  "  whiche  doth  not  onely  serve  for  them  that  cannot 
write  and  reade,  but  also  for  them  that  can  doe  both,  but  have 
not  at  some  times  their  pen  or  tables  readie  with  them."  He 
mentions  two  ways  of  representing  sums  by  counters,  the  Mer- 
chant's and  the  Auditor's  account.  In  the  first,  198  1.,  19  s., 
11  d.,  is  expressed  by  counters  (our  dots)  thus :  — 

•  •     •         =100  +  80  pounds. 

•  '  =10+5  +  3  pounds. 

•  •     •         =10  +  5  +  4  shillings. 

•  '     *         =6  +  5  pence. 


186  A   HISTORY   OF   MATHEMATICS 

Observe  that  the  four  horizontal  lines  stand  respectively  for 
pence,  shillings,  pounds,  and  scores  of  pounds,  that  counters  in 
the  intervening  spaces  denote  half  the  units  in  the  next  line 
above,  and  that  the  detached  counters  to  the  left  are  equivalent 
to  five  counters  to  the  right.1 

The  abacus  with  its  counters  had  ceased  to  be  used  in  Spain 
and  Italy  in  the  fifteenth  century.  In  France  it  was  used  at 
the  time  of  Eecorde,  and  it  did  not  disappear  in  England  and 
Germany  before  the  middle  of  the  seventeenth  century.  The 
method  of  abacal  computation  is  found  in  the  English  ex- 
chequer for  the  last  time  in  1676.  In  the  reign  of  Henry  I. 
the  exchequer  was  distinctly  organized  as  a  court  of  law,  but 
the  financial  business  of  the  crown  was  also  carried  on  there. 
The  term  "exchequer77  is  derived  from  the  chequered  cloth 
which  covered  the  table  at  which  the  accounts  were  made  up. 
Suppose  the  sheriff  was  summoned  to  answer  for  the  full  annual 
dues  "  in  money  or  in  tallies."  "  The  liabilities  and  the  actual 
payments  of  the  sheriff  were  balanced  by  means  of  counters 
placed  upon  the  squares  of  the  chequered  table,  those  on  the 
one  side  of  the  table  representing  the  value  of  the  tallies, 
warrants,  and  specie  presented  by  the  sheriff,  and  those  on  the 
other  the  amount  for  which  he  was  liable,"  so  that  it  was  easy 
to  see  whether  the  sheriff  had  met  his  obligations  or  not.  In 
Tudor  times  "  pen  and  ink  dots  "  took  the  place  of  counters. 
These  dots  were  used  as  late  as  1676.2  The  "tally"  upon 
which  accounts  were  kept  was  a  peeled  wooden  rod  split  in 
such  a  way  as  to  divide  certain  notches  previously  cut  in  it. 
One  piece  of  the  tally  was  given  to  the  payer ;  the  other  piece 
was  kept  by  the  exchequer.  The  transaction  could  be  verified 
easily  by  fitting  the  two  halves  together  and  noticing  whether 

1  PEACOCK,  p.  410 ;  Peacock  also  explains  the  Auditor's  Account. 

2  Article  "Exchequer"  in  PALGRAVB'S  Dictionary  of  Political  Econ- 
omy, London,  1894. 


MODERN    TIMES  187 

the  notches  "  tallied  "  or  not.  Such  tallies  remained  in  use  as 
late  as  1783.1 

In  the  Winters  Tale  (IV.,  3),  Shakespeare  lets  the  clown  be 
embarrassed  by  a  problem  which  he  could  not  do  without 
counters.  lago  (in  Othello,  I.)  expresses  his  contempt  for 
Michael  Cassio,  "  forsooth  a  great  mathematician,"  by  calling 
him  a  "  counter-caster.7' 2  It  thus  appears  that  the  old  methods 
of  computation  were  used  long  after  the  Hindu  numerals  were 
in  common  and  general  use.  With  such  dogged  persistency 
does  man  cling  to  the  old ! 

While  England,  during  the  sixteenth  century,  produced  no 
mathematicians  comparable  with  Vieta  in  France,  Rheticus 
in  Germany,  or  Cataldi  in  Italy,  it  is  nevertheless  true  that 
Tonstall  and  Recorde,  through  their  mathematical  works,  re- 
flect credit  upon  England.  Their  arithmetics  are  above  the 
average  of  European  works.  With  the  time  of  Recorde  the 
English  began  to  excel  in  numerical  skill  as  applied  to  money. 
"  The  questions  of  the  English  books,"  says  De  Morgan,  "  are 
harder,  involve  more  figures  in  data,  and  are  more  skilfully 
solved."  To  this  fact,  no  doubt,  we  must  attribute  the  ready 
appreciation  of  decimal  fractions,  and  the  instantaneous  popu- 
larity of  logarithms.  The  number  of  arithmetical  writers  in 
the  seventeenth  and  eighteenth  centuries  is  very  large.  Among 
the  more  prominent  of  the  early  writers  after  Tonstall  and 
Recorde  are : 3  William  Buckley,  mathematical  tutor  of 
Edward  VI.  and  author  of  Arithmetica  Memorativa  (1550) ; 
Humfrey  Baker,  author  of  The  Well-Spring  of  the  Sciences 
(1562) ;  Edmund  Wingate,  whose  Arithmetic^  appeared  about 
1629;  William  Oughtred,  who  in  1631  published  his  Clavis 

1  "Tallies  have  been  at  6,  but  now  at  5  per  cent,  per  annum,  the  in- 
terest payable  every  three  months."     SHELLEY'S  WINGATE'S  Arithmetic, 
1735,  p.  407. 

2  PEACOCK,  p.  408.  »  PEACOCK,  pp.  437,  441,  442,  452. 


188  A   HISTORY   OF   MATHEMATICS 

Mathematicce,  a  systematic  text-book  on  arithmetic  and  algebra ; 
Noah  Bridges,  author  of  Vulgar  Arithmeticke  (1653) ;  Andrew 
Tacquet,  a  Jesuit  mathematician  of  Antwerp,  author  of  several 
books,  in  particular  of  Arithmetics  Theoria  et  Praxis  (Antwerp, 
1656,  later  reprinted  in  London).  Mention  should  be  made 
also  of  The  Pathway  of  Knowledge,  an  anonymous  work,  written 
in  Dutch  and  translated  into  English  in  1596.  John  Mellis, 
in  1588,  issued  the  first  English  work  on  book-keeping  by 
double  entry.1 

We  have  seen  that  the  invention  of  printing  revealed  in 
Europe  the  existence  of  two  schools  of  arithmeticians,  the 
algoristic  school,  teaching  rules  of  computation  and  commercial 
arithmetic,  and  the  abacistic  school,  which  gave  no  rules  of 
calculation,  but  studied  the  properties  of  numbers  and  ratios. 
Boethius  was  their  great  master,  while  the  former  school 
followed  in  the  footsteps  of  the  Arabs.  The  algoristic  school 
flourished  in  Italy  (Pacioli,  Tartaglia,  etc.)  and  found  adher- 
'ents  throughout  England  and  the  continent.  But  it  is  a 
remarkable  fact  that  the  abacistic  school,  with  its  pedantry, 
though  still  existing  on  the  continent,  received  hardly  any 
attention  in  England.  The  laborious  treatment  of  arithmetical 
ratios  with  its  burdensome  phraseology  was  of  no  practical 
use  to  the  English  merchant.  The  English  mind  instinctively 
rebelled  against  calling  the  ratio  3:2  =  1^-  by  the  name  of 
proportio  superparticularis  sesquialtera. 

On  the  other  hand  it  is  a  source  of  regret  that  the  successors 
of  Tonstall  and  Recorde  did  not  observe  the  high  standard  of 
authorship  set  by  these  two  pioneers.  A  decided  decline  is 
marked  by  Buckley's  Arithmetica  Memorativa,  a  Latin  treatise 
expressing  the  rules  of  arithmetic  in  verse,  which,  we  take  it, 
was  intended  to  be  committed  to  memory.  We  are  glad  to  be 

1  DE  MORGAN,  Arith.  Books,  p.  27. 


MODERN   TIMES  189 

able  to  say  that  this  work  never  became  widely  popular.  The 
practice  of  expressing  rules  in  verse  was  common,  before  the 
invention  of  printing,  but  the  practice  of  using  them  was  not 
common,  else  the  number  of  printed  arithmetics  on  this  plan 
would  have  been  much  larger  than  it  actually  was.1  Many 
old  arithmetics  give  occasionally  a  rhyming  rule,  but  few  con- 
fine themselves  to  verse.  Few  authors  are  guilty  of  the  folly 
displayed  by  Buckley  or  by  Solomon  Lowe,  who  set  forth 
the  rules  of  arithmetic  in  English  hexameter,  and  in  alpha- 
betical order. 

In  this  connection  it  may  be  stated  that  an  early  specimen 
of  the  muse  of  arithmetic,  first  found  in  the  Pathway  of  Know- 
ledge, 1596,  has  come  down  to  the  present  generation  as  the 
most  classical  verse  of  its  kind: 

"Thirtie  dales  hath  September,  Aprill,  June,  and  November, 
Februarie  eight  and  twentie  alone,  all  the  rest  thirtie  and  one." 

As  a  close  competitor  for  popularity  is  the  following  stanza 2 
quoted  by  Mr.  Davies  (Key  to  Hutton's  "  Course ")  from  a 
manuscript  of  the  date  1570  or  near  it: 

**  Multiplication  is  mie  vexation 
And  Division  is  quite  as  bad, 
The  Golden  Kule  is  mie  stumbling  stule 
And  Practice  drives  me  mad." 

In  the  sixteenth  century  instances  occur  of  arithmetics  writ- 
ten in  the  form  of  questions  and  answers.  During  the  seven- 
teenth century  this  practice  became  quite  prevalent  both  in 
England  and  Germany.  We  are  inclined  to  agree  with 
Wildermuth  that  it  is,  on  the  whole,  an  improvement  on  the 
older  practice  of  simply  directing  the  student  to  do  so  and  so. 

1  DE  MORGAN,  Arith.  Books,  p.  16. 

2  DE  MORGAN,  Arith.  Books. 


190  A   HISTORY   OF   MATHEMATICS 

A  question  draws  the  pupil's  attention  and  prepares  his  mind 
for  the  reception  of  the  new  information.  Unfortunately,  the 
question  always  relates  to  how  a  thing  is  done,  never  why  it  is 
done  as  indicated.  It  is  deplorable  to  see  in  the  seventeenth 
century,  both  in  England  and  Germany,  that  arithmetic  is 
reduced  more  and  more  to  a  barren  collection  of  rules.  The 
sixteenth  century  brought  forth  some  arithmetics,  by  promi- 
nent mathematicians,  in  which  attempts  were  made  at  demon- 
stration. Then  follows  a  period  in  which  arithmetic  was 
studied  solely  for  commercial  purposes,  and  to  this  commercial 
school  of  arithmeticians  (about  the  middle  of  the  seventeenth 
century),  says  De  Morgan,1  "  we  owe  the  destruction  of  demon- 
strative arithmetic  in  this  country,  or  rather  the  prevention 
of  its  growth.  It  never  was  much  the  habit  of  arithmeticians 
to  prove  their  rules ;  and  the  very  word  proof,  in  that  science, 
never  came  to  mean  more  than  a  test  of  the  correctness  of  a 
particular  operation,  by  reversing  the  process,  casting  out  the 
nines,  or  the  like.  As  soon  as  attention  was  fairly  averted  to 
arithmetic  for  commercial  purposes  alone,  such  rational  appli- 
cation as  had  been  handed  down  from  the  writers  of  the  six- 
teenth century  began  to  disappear,  and  was  finally  extinct  in 
the  work  of  Cocker  or  Hawkins,  as  I  think  I  have  shown  rea- 
son for  supposing  it  should  be  called.2  From  this  time  began 
the  finished  school  of  teachers,  whose  pupils  ask,  when  a  ques- 
tion is  given,  what  rule  it  is  in,  and  run  away,  when  they  grow 
up,  from  any  numerical  statement,  with  the  declaration  that 

1  Arith.  Books,  p.  21. 

2  "  Cocker's  Arithmetic  "  was  "  perused  and  published  "  after  Cocker's 
death  by  John  Hawkins.     De  Morgan  claims  that  the  work  was  not  writ- 
ten by  Cocker  at  all,  but  by  John  Hawkins,  and  that  Hawkins  attached 
to  it  Cocker's  name  to  make  it  sell.     After  reading  the  article  "  Cocker" 
in  the  Dictionary  of  National  Biography,  we  are  confident  in  believing 
Hawkins  innocent.      Cocker's  sudden  death  at  an  early  age  is  sufficient 
to  account  for  most  of  his  works  being  left  for  posthumous  publication. 


MODERN   TIMES  191 

anything  may  be  proved  by  figures  —  as  it  may,  to  them) 
Anything  may  be  unanswerably  propounded,  by  means  of 
figures,  to  those  who  cannot  think  upon  numbers.  Towards 
the  end  of  the  last  century  we  see  a  succession  of  works, 
arising  one  after  the  other,  all  complaining  of  the  state  into 
which  arithmetic  had  fallen,  all  professing  to  give  rational 
explanation,  and  hardly  one  making  a  single  step  in  advance 
of  its  predecessors. 

"  It  may  very  well  be  doubted  whether  the  earlier  arithme- 
ticians could  have  given  general  demonstrations  of  their  pro- 
cesses. It  is  an  unquestionable  fact  of  observation  that  the 
application  of  elementary  principles  to  their  apparently  most 
natural  deduction,  without  drawing  upon  subsequent,  or  what 
ought  to  be  subsequent,  combinations,  seldom  takes  place  at 
the  commencement  of  any  branch  of  science.  It  is  the  work 
of  advanced  thought.  But  the  earlier  arithmeticians  and  alge- 
braists had  another  difficulty  to  contend  with:  their  fear  of 
their  own  half -under  stood  conclusions,  and  the  caution  with 
which  it  obliged  them  to  proceed  in  extending  their  half- 
formed  language." 

Of  arithmetical  authors  belonging  to  the  commercial  school 
we  mention  (besides  Cocker)  James  Hodder,  Thomas  Dilworth, 
and  Daniel  Penning,  because  we  shall  find  later  that  their 
books  were  used  in  the  American  colonies. 

James  Hodder2  was  in  1661  writing-master  in  Lothbury, 
London.  He  was  first  known  as  the  author  of  Hodder's 
Arithmetick,  a  popular  manual  upon  which  Cocker  based  his 
better  known  work.  Cocker's  chief  improvement  is  the  use 

1  The  period  of  teaching  arithmetic  wholly  by  rules  began  in  Germany 
about  the  middle  of  the  sixteenth  century  —  nearly  one  hundred  years 
earlier  than  in  England.     But  the  Germans  returned  to  demonstrative 
arithmetic  in  the  eighteenth  century,  at  an  earlier  period  than  did  the 
British.     See  UNGER,  pp.   iv,  117,  137. 

2  Dictionary  of  National  Biography. 


192  A    HISTORY   OF   MATHEMATICS 

of  the  new  mode  of  division,  "by  giving"  (as  the  Italians 
called  it),  in  place  of  the  "  scratch "  or  "  galley "  method 
taught  by  Hodder.  The  first  edition  of  Hodder's  appeared 
in  1661,  the  twentieth  edition  in  1739.  He  wrote  also  The 
Penman's  Recreation  (the  specimens  of  which  are  engraved 
by  Cocker,  with  whom,  it  appears,  Hodder  was  friendly), 
and  Decimal  Arithmetick,  1668. 

Cocker's  Arithmetic^  went  through  at  least  112  editions,1 
including  Scotch  and  Irish  editions.2  Like  Franqois  Barreme 
in  France  and  Adam  Riese  in  Germany,  Cocker  in  England 
enjoyed  for  nearly  a  century  a  proverbial  celebrity,  these  names 
being  synonymous  with  the  science  of  numbers.  A  man  who 
influenced  mathematical  teaching  to  such  an  extent  deserves 
at  least  a  brief  notice.  Edward  Cocker  (1631-1675)  was  a  prac- 
titioner in  the  arts  of  writing,  arithmetic,  and  engraving.  In 
1657  he  lived  "  on  the  south  side  of  St.  Paul's  Churchyard " 
where  he  taught  writing  and  arithmetic  "  in  an  extraordinary 
manner."  In  1664  he  advertised  that  he  would  open  a  public 
school  for  writing  and  arithmetic  and  take  in  boarders  near 
St.  Paul's.  Later  he  settled  at  Northampton'.3  Aside  from 

1  Dictionary  of  National  Biography. 

2  "  Corrected "  editions  of  Cocker's  Arithmetic  were  brought  out  in 
1725,  1731, 1736, 1738, 1745, 1758,  1767  by  "  George  Fisher,"  a  pseudonym 
for  MRS.  SLACK.     Under  the  same  pseudonym  she  published  in  London, 
1763,  The  Instructor :  or  Young  Man's  best  Companion,  containing  spell- 
ing, reading,  writing,  and  arithmetic,  etc.,  the  fourteenth  edition  of  which 
appeared  in  1785  at  Worcester,  Mass.,  (the  twenty-eighth  in  1798)  under 
the  title,  The  American  Instructor :  etc.,  as  above.     In  Philadelphia  the 
book  was  printed  in  1748  and  1801.     Mrs.  Slack  is  the  first  woman  whom 
we  have  found  engaged  in  arithmetical   authorship.     See   Bibliotheca 
Mathematica,  1895,  p.  75  ;   Teach,  and  Hist,  of  Mathematics  in  the  U.  S., 
1890,  p.  12. 

3  PEPYS  mentions  him   several  times  in   1664   in  his   Diary,   10th : 
"Abroad  to  find  out  one  to  engrave  my  table  upon  my  new  sliding  rule 
with  silver  plates,  it  being  so  small  that  Browne  who  made  it  cannot  get 
one  to  do  it.     So  I  got  Cocker,  the  famous  writing  master,  to  do  it,  and 


MODERN   TIMES  193 

the  absence  of  all  demonstration,  Cocker's  Arithmetic  was  well 
written,  and  it  evidently  suited  the  demands  of  the  times.  He 
was  a  voluminous  writer,  being  the  author  of  33  works,  23 
calligraphic,  6  arithmetical,  and  4  miscellaneous.  The  arith- 
metical books  are :  Tutor  to  Arithmetic!^  (1664) ;  Compleat 
Arithmetician  (1669)  ;  Arithmetic^,  (1678 ;  Peacock  on  page  454 
gives  the  date  1677) ;  Decimal  Arithmetic!*;,  Artificial  Aritli- 
metick (being  of  logarithms),  Algebraical  Arithmetic^  (treating 
of  equations)  in  three  parts,  1684,  1685,  "perused  and  pub- 
lished" by  John  Hawkins. 

In  the  English,  as  well  as  the  French  and  German  arith- 
metics, which  appeared  during  the  sixteenth,  seventeenth,  and 
eighteenth  centuries,  the  "rule  of  three"  occupies  a  central 
position.  Baker  says 1  in  his  Well-Spring  of  the  Sciences,  1562, 
"The  rule  of  three  is  the  chief est,  and  the  most  profitable, 
and  most  excellent  rule  of  all  arithmeticke.  For  all  other 
rules  have  neede  of  it,  and  it  passeth  all  other ;  for  the  which 
cause,  it  is  sayde  the  philosophers  did  name  it  the  Golden 
Rule,  but  now,  in  these  later  days,  it  is  called  by  us  the  Eule 
of  Three,  because  it  requireth  three  numbers  in  the  operation." 
There  has  been  among  writers  considerable  diversity  in  the 
notation  for  this  rule.  Peacock  (p.  452)  states  the  question 
"  If  2  applet  cost  3  soldi,  what  will  13  cost  ?  "  and  with  refer- 
ence to  it,  represents  Tartaglia's  notation  thus, 

Se  pomi  2  ||  val  soldi  3  ||  che  valera  pomi  13. 

I  set  an  hour  by  him  to  see  him  design  it  all ;  and  strange  it  is  to  see 
him  with  his  natural  eyes  to  cut  so  small  at  his  first  designing  it,  and  read 
it  all  over,  without  any  missing,  when  for  my  life  I  could  not,  with  my 
best  skill,  read  one  word,  or  letter  of  it.  ...     I  find  the  fellow  by  his  dis- 
course very  ingenious  ;  and  among  other  things,  a  great  admirer  and  well 
read  in  the  English  poets,  and  undertakes  to  judge  of  them  all,  and  that 
not  impertinently."    Cocker  wrote  quaint  poems  and  distichs  which  show 
some  poetical  ability. 
1  PEACOCK,  p.  452. 
o 


194  A   HISTORY   OF   MATHEMATICS 

Recorde    and    the    older   English   arithmeticians   write   as 

follows  : 

Apples.      Pence. 

2^—-    3 
13   ^19    answer. 


In  the  seventeenth  century  the  custom  was  as  follows  (Win- 
gate,  Cocker,  etc.)  : 

Apples.    Pence.     Apples. 
2  -  3  -  13 

The  notation  of  this  subject  received  the  special  attention  of 
Oughtred,  who  introduced  the  sign  :  :  and  wrote 


M.  Cantor l  says  that  the  dot,  used  here  to  express  the  ratio, 
later  yielded  to  two  dots,  for  .in  the  eighteenth  century  the 
German  writer,  Christian  Wolf,  secured  the  adoption  of  the  dot 
as  the  usual  symbol  of  multiplication.  In  England  the  reason 
for  the  change  from  dot  to  colon  was  a  different  one.  It  will 
be  remembered  that  Oughtred  did  not  use  the  decimal  point. 
Its  general  introduction  in  England  took  place  in  the  first 
quarter  of  the  eighteenth  century,  and  we  are  quite  sure  that 
it  was  the  decimal  point  and  not  Wolf's  multiplication  sign 
which  displaced  Oughtred's  symbol  for  ratio.  As  the  Ger- 
mans use  a  decimal  comma 2  instead  of  our  point,  the  reason 

1  CANTOR,  II.,  p.  721. 

2  The   Germans  attribute  the  introduction  of  the  decimal  comma  to 
Kepler  (see  UNGER,  p.  104 ;  GERHARDT,  pp.  78, 109;  GUNTHER,  Vermischte 
Untersuchungen,  p.  133).     Reuses  it  in  a  publication  of  1616.     Napier, 
in  his  Rabdologia  (1617)  speaks  of  "adding  a  period  or  comma,"  and 
writes  1993,273  (see  Construction,  MACDONALD'S  Ed.,  p.  89).     English 
writers  did  not  confine  themselves  to  the  decimal  point;  the  comma  is 
often  used.     Thus,  Martin's  Decimal  Arithmetick  (1763)  and  Wilder' s 
edition    of    Newton's    Universal    Arithmetick,    London,    1769,   use   the 
comma  exclusively.     In  Kersey's  Wingate  (1735)  and  in  Dilworth  (1784), 


MODERN   TIMES  195 

for  the  change  could  not  have  been  the  same  in  Germany  as 
in  England.  Dilworth l  does  not  use  the  dot  in  multiplication, 
but  he  employs  the  decimal  point  and  writes2  proportion 
once  2  •  •  4  : :  8  •  •  16,  and  on  another  page  3  : 17  : :  48.  Before 
the  present  century,  the  dot  was  seldom  used  by  English 
writers  to  denote  multiplication.  If  Oughtred  had  been  in 
the  habit  of  regarding  a  proportion  as  the  equality  of  two 
ratios,  then  he  would  probably  have  chosen  the  symbol  = 
instead  of  :  : .  The  former  sign  was  actually  used  for  this 
purpose  by  Leibniz.3  The  notation  2:4  =  1:2  was  brought 
into  use  in  the  United  States  and  England  during  the  first 
quarter  of  the  nineteenth  century,  when  Euler's  Algebra  and 
French  text-books  began  to  be  studied  by  the  English-speaking 
nations. 

The  rule  of  three  reigned  supreme  in  commercial  arith- 
metics in  Germany  until  the  close  of  the  eighteenth  century, 
and  in  England  and  America  until  the  close  of  the  first  quarter 
of  the  present  century.4  It  has  been  much  used  since.  An 

we  read  of  the  "point  or  comma,"  but  the  point  is  the  sign  actually 
used.  In  Dodson's  Wingate  (1760)  both  the  comma  and  point  are  used, 
the  latter,  perhaps,  more  frequently.  Cocker  (1714)  and  Hatton  (1721) 
do  not  even  mention  the  comma. 

1  THOMAS  DILWORTH,    Schoolmaster's   Assistant,   22d  Ed.,   London, 
1784,  page  after  table  of  contents ;  also,  pp.  45,  123.    The  earliest  testi- 
monials are  dated  1743. 

2  In  his  time  old  and  new  notations  were  in  simultaneous  use,  for  he 
says,   "Some  masters,  instead  of  points,  use  long  strokes  to  keep  the 
terms  separate,  but  it  is  wrong  to  do  so ;  for  the  two  points  between  the 
first  and  second  terms,  and  also  between  the  third  and  fourth  terms, 
shew  that  the  two  first  and  the  two  last  terms  are  in  the  same  proportion. 
And  whereas  four  points  are  put  between  the  second  and  third  terms, 
they  serve  to  disjoint  them,  and  shew  that  the  second  and  third,  and 
first  and  fourth  terms  are  not  in  the  same  direct  proportion  to  each 
other  as  are  these  before  mentioned."  3  WILDERMUTH. 

*  UNGER  (p.  170)  says  that  in  Germany  the  rule  of  three  was  preferred 
as  the  universal  rule  for  problem-working,  during  the  sixteenth  century; 


196  A   HISTORY   OF   MATHEMATICS 

important  role  was  played  in  commercial  circles  by  an  allied 
rule  called  in  English  chain-rule  or  conjoined  proportion,  in 
French  regie  conjointe,  and  in  German  Kettensatz  or  Reesischer 
Satz.1  It  received  its  most  perfect  formal  development  and 
most  extended  application  in  Germany  and  the  Netherlands. 
Kelly2  attributes  the  superiority  of  foreign  merchants  in  the 
science  of  exchange  to  a  more  intimate  knowledge  of  this  rule. 
The  chain-rule  was  known  in  its  essential  feature  to  the  Hindu 
Brahmagupta ;  also  to  the  Italians,  Leonardo  of  Pisa,  Pacioli, 
Tartaglia ;  to  the  early  Germans,  Johann  Widmann  and  Adam 
Biese.  In  England  the  rule  was  elaborated  by  John  Kersey 
in  his  edition  of  1668  of  Wingate's  arithmetic.  But  the  one 
who  contributed  most  toward  its  diffusion  was  Kaspar  Franz 
von  Eees  (born  1690)  of  Eoermonde  in  Limburg,  who  migrated 
to  Holland.  There  he  published  in  Dutch  an  arithmetic  which 
appeared  in  French  translation  in  1737,  and  in  German  transla- 
tion in  1739.  In  Germany  the  ^Reesischer  Satz  became  famous. 
Cocker  (28th  Ed.,  Dublin,  1714,  p.  232)  illustrates  the  rule  by 
the  example :  "  If  40  1.  Auverdupois  weight  at  London  is  equal 
to  36  1.  weight  at  Amsterdam,  and  90  1.  at  Amsterdam  makes 
116  1.  at  Dantzick,  then  how  many  Pounds  at  Dantzick  are  equal 
to  112  1.  Auverdupois  weight  at  London  f  " 

the  method,  called  Practice,  being  preferred  during  the  seventeenth  cen- 
tury, the  Chain-Rule  during  the  eighteenth,  and  Analysis  (Bruchsatz,  or 
Schlussrechnung)  during  the  nineteenth.  We  fail  to  observe  similar 
changes  in  England.  There  the  rule  of  three  occupied  a  commanding 
position  until  the  present  century  ;  the  rules  of  single  arid  double  position 
were  used  in  the  time  of  Recorde  more  than  afterwards ;  the  Chain-Rule 
never  became  widely  popular ;  while  some  attention  was  always  paid  to 
Practice. 

1  The  writer  remembers  being  taught  the  «« Kettensatz  "  in  1873  at  the 
Kantonsschule  in  Chur,  Switzerland,  along  with   three   other  methods, 
the  "Einheitsmethode"  (Schlussrechnung),  the  "  Zerlegungsmethode  " 
(a  form  of  Italian  practice)  and  "  Proportion." 

2  Vol.  II.,  D.  3. 


MODERN   TIMES  197 

(p.  234)  "  The  Terms  being  disposed  according  to  the  7th 
Rule  foregoing,  will  stand  thus, 


A      B 

1.  at  Lond. 

40 

36 

1.  at  Amsterdam 

1.  at  Amst. 

90 

116 

1.  at  Dantzick 

112 

1.  at  London 

whereby  I  find  that  the  terms  under  B  multiplied  together  pro- 
duce 467712  for  a  Dividend  and  the  terms  under  A,  viz.,  40  and 
90  produce  3600  for  a  Divisor,  and  Division  being  finished,  the 
Quotient  giveth  129Jf^J  Pounds  at  Dantzick  for  the  Answer." 
The  chain-rule  owes  its  celebrity  to  the  fact  that  the  correct 
answer  could  be  obtained  without  any  exercise  of  the  mind. 
Rees  reduced  the  statement  to  a  mere  mechanism.  While 
useful  to  the  merchant,  the  rule  was  worthless  for  mind-culture 
in  the  school-room.  Attempts  were  made  by  some  arithmetical 
writers  to  prove  it  with  aid  of  proportions  or  by  ordinary 
analysis.  But  for  pedagogical  purposes  those  attempts  proved 
unsatisfactory.  A  rule,  more  apt  to  lead  to  errors,  but  requir- 
ing some  thought,  became  known  in  Germany  as  "  Basedowsche 
Regel,"  being  recommended  by  the  educational  reformer  Base- 
dow,  though  not  first  given  by  him.  Early  in  the  present 
century,  arithmetical  teaching  was  revolutionized  in  Germany. 
The  chain-rule  and  rule  of  three  were  gradually  driven  into  the 
background,  and  the  "  Schlussrechnung "  attained  more  and 
more  prominence,  even  though  Pestalozzi  was  partial  to  the 
use  of  proportion.  The  "Schlussrechnung"  is  sometimes 
designated  in  English  by  the  word  Analysis.  If  3  yards  cost 
$  7,  what  will  19  yards  cost  ?  One  yard  will  cost  $  J,  and  19 

yards  ®7  X  19  =  $  44.33;  or  by  a  modification  of  the  above,  by 
3 

"aliquot  parts":   18  yards  will  cost  $42;   one  yard  $2.33, 
and  19  yards,  $  44.33.     This  "  Schlussrechnung  "  was  known 


198  A   HISTORY   OF  MATHEMATICS 

to  Tartaglia,  but  is,  without  doubt,  very  much  older ;  for  it  is 
a  thoroughly  natural  method  which  would  suggest  itself  to  any 
sound  and  vigorous  mind.  Its  pedagogical  value  lies  in  the 
fact  that  there  is  no  mechanism  about  it,  that  it  requires  no 
memorizing  of  formulae,  but  makes  arithmetic  an  exercise  in 
thinking.  It  is  strange  indeed  that  in  modern  times  such  a 
method  should  have  been  disregarded  by  arithmeticians  dur- 
ing three  long  centuries. 

English  arithmetics  embraced  all  the  commercial  subjects 
previously  used  by  the  Italians,  such  as  simple  and  compound 
interest,  the  direct  rule  of  three,  the  inverse  rule  of  three 
(called  by  Recorde  "backer  rule  of  three"),  loss  and  gain, 
barter,  equation  of  payments,  bills  of  exchange,  alligation, 
annuities,  the  rules  for  single  and  double  position,  and  the 
subject  of  tare,  trett,  cloff.  We  let  Dil worth  (p.  37,  1784) 
explain  the  last  three  terms: 

"  Q.  Which  are  the  Allowances  usually  made  in  Averdupois  great 
Weight  to  the  Buyer? 

A.   They  are  Tare,  Trett,  and  Cloff. 

Q.    What  is  Tare? 

A.  Tare  is  an  Allowance  made  to  the  Buyer,  for  the  Weight  of  the 
Box,  Bag,  Vessel,  or  whatever  else  contains  the  Goods  bought.  .  .  . 

Q.    What  is  Trett  ? 

A.  Trett  is  an  Allowance  by  the  Merchant  to  the  Buyer  of  4  Ib.  in 
104  Ib.,  that  is,  the  six-and-twentieth  Part,  for  Waste  or  Dust,  in  some 
Sorts  of  Goods.  .  .  . 

Q.   What  is  Cloff? 

A.  Cloff  is  an  Allowance  of  2  Ib.  Weight  to  the  Citizens  of  London, 
on  every  Draught  above  3  C.  Weight,  on  some  Sorts  of  Goods,  as  Galls, 
Madder,  Sumac,  Argol,  etc.1 

Q.    What  are  these  Allowances  called  beyond  the  Seas  ? 

A.  They  are  called  the  Courtesies  of  London  /  because  they  are  not 
practiced  in  any  other  Place." 

1  The  term  "  cloff  "  had  also  a  more  general  meaning,  denoting  a  small 
allowance  made  on  goods  sold  in  gross,  to  make  up  for  deficiences  in 
weight  when  sold  in  retail.  PEACOCK,  p.  455. 


MODERN   TIMES  199 

To  the  above  subjects,  which  were  borrowed  from  the 
Italians,  English  writers  added  their  own  weights  and  meas- 
ures, and  those  of  the  countries  with  which  England  traded. 
In  the  seventeenth  century  were  added  decimal  fractions,  which 
were  taught  in  books  more  assiduously  then  than  a  century 
later.  It  is  surprising  to  find  that  some  arithmetics  devoted 
considerable  attention  to  logarithms.  Cocker  wrote  a  book  on 
"Artificial  Arithmetick."  We  have  seen  the  rules  for  the 
use  of  logarithms,  together  with  logarithmic  tables  of  num- 
bers, in  the  arithmetics  of  John  Hill  (10th  Ed.  by  E.  Hatton, 
London,  1761),  Benjamin  Martin  (Decimal  Arithmetick,  London, 
1763;  said  to  have  been  first  published  in  1735),  Edward 
Hatton  (Intire  System  of  Arithmetic,  London,  1721)  and  in 
editions  of  Edmund  Wingate.  Wingate  was  a  London  lawyer 
who  pursued  mathematics  for  pastime.  Spending  a  few  years 
in  Paris,  he  published  there  in  1625  *  his  Arithmetique  loga- 
rithmique,  which  appeared  in  London  in  English  translation 
in  1635.2  Wingate  was  the  first  to  carry  Briggian  logarithms 
(taken  from  Gunter's  tables)  into  France.  About  the  year 
1629  he  published  his  Arithmetick  "in  which  his  principal 
design  was  to  obviate  the  difficulties  which  ordinarily  occur 
in  the  using  of  logarithms :  To  perform  this  he  divided  his 
work  into  two  books;  the  first  he  called  Natural,  and  the 
second  Artificial  Arithmetic" 3  Subsequent  editions  of  Wingate 
rested  on  the  first  of  those  two  books,  and  were  brought  out 
by  John  Kersey,  later  by  George  Shelley,  and  finally  by  James 
Dodson.  The  work  was  modified  so  largely,  that  Wingate 
would  not  have  known  it. 

1  DE  MORGAN,  Arith.  Books;  In  MAXIMILIEN  MARIE'S,  Histoire  des 
Sciences  Mathematiques  et  Physiques,  Vol.  III.,  p.  225,  is  given  the  date 
1626,  instead  of  1625.  2  MARIE,  Vol.  III.,  p.  225. 

8  JAMES  DODSON'S  Preface  to  WINGATE'S  Arithmetic,  London,  1760. 
The  term  "  artificial  numbers  "  for  logarithms  is  due  to  Napier  himself. 


200  A    HISTORY   OF   MATHEMATICS 

It  is  interesting  to  observe  how  writers  sometimes  intro- 
duced subjects  of  purely  theoretical  value  into  practical  arith- 
metics. Perhaps  authors  thought  that  theoretical  points 
which  they  themselves  had  mastered  and  found  of  interest, 
ought  to  excite  the  curiosity  of  their  readers.  Among  these 
subjects  are  square-,  cube-,  and  higher  roots,  continued  frac- 
tions, circulating  decimals,  and  tables  of  the  powers  of  2  up  to 
the  144th.  The  last  were  "  very  useful  for  laying  up  grains  of 
corn  on  the  squares  of  a  chess-board,  ruining  people  by  horse- 
shoe bargains,  and  other  approved  problems"  (De  Morgan). 
The  subject  of  circulating  decimals  was  first  elaborated  by 
John  Wallis  (Algebra,  Ch.  89),  Leonhard  Euler  (Algebra,  Book 
L,  Ch.  12),  and  John  Bernoulli.  Circulating  decimals  were  at 
one  time  "  suffered  to  embarrass  books  on  practical  arithmetic, 
which  need  have  no  more  to  do  with  them  than  books  on 
mensuration  with  the  complete  quadrature  of  the  circle." ] 
The  Decimal  Arithmetic  of  1742  by  John  Marsh  is  almost 
entirely  on  this  subject.  As  his  predecessors  he  mentions 
Wallis,  Jones  (1706),  Ward,  Brown,  Malcolm,  Cunn,  Wright. 

After  the  great  fire  of  London,  in  1666,  the  business  of  fire 
insurance  began  to  take  practical  shape,  and  in  1681  the  first 
regular  fire  insurance  office  was  opened  in  London.  The  first 
office  in  Scotland  was  established  in  1720,  the  first  in  Ger- 
many in  1750,  the  first  in  the  United  States  at  Philadelphia 
in  1752,  with  Benjamin  Franklin  as  one  of  the  directors.2 
In  course  of  time,  fire  insurance  received  some  attention  in 
English  arithmetics.  In  1734  the  first  approach  to  modern 
life  insurance  was  made,  but  all  members  were  rated  alike, 
irrespective  of  age.  In  1807  we  have  the  first  instance  of 
rating  "according  to  age  and  other  circumstances." 

It  is  interesting  to  observe  that  before  the  middle  of  the 

1  DE  MORGAN,  Arith.  Books,  p.  69. 

2  Article  "  Insurance  "  in  the  Encyclopedia  Britannica,  9th  Ed. 


MODERN   TIMES  201 

eighteenth  century  it  was  the  custom  in  England  to  begin 
the  legal  year  with  the  25th  of  March.  Not  until  1752 
did  the  counting  of  the  new  year  begin  with  the  first  of 
January  and  according  to  the  Gregorian  calendar.1  In  1752, 
eleven  days  were  dropped  between  the  2d  and  14th  of 
September,  thereby  changing  from  "old  style"  to  "new 
style." 

The  order  in  which  the  various  subjects  were  treated  in  old 
arithmetics  was  anything  but  logical.  The  definitions  are 
frequently  given  in  a  collection  at  the  beginning.  Dilworth 
develops  the  rules  for  "whole  numbers,"  then  develops  the 
same  rules  for  "vulgar  fractions,"  and  again  for  "decimal 
fractions."  Thus  he  gives  the  "  rule  of  three,"  later  the  "  rule 
of  three  in  fractions,"  and,  again,  the  "  rule  of  three  direct  in 
decimals."  John  Hill,  in  his  arithmetic  (edition  of  1761),  adds 
to  this  "  the  golden  rule  in  logarithms."  Fractions  are  taken 
up  late.  Evidently  many  students  had  no  expectation  of  ever 
reaching  fractions,  and,  for  their  benefit,  the  first  part  of  the 
arithmetics  embraced  all  the  commercial  rules.  In  the  eigh- 
teenth century  the  practice  of  postponing  fractions  to  the 
last  became  more  prevalent.2  Moreover,  the  treatment  of 
this  subject  was  usually  very  meagre.  While  the  better  types 

1  E.  STONE  in  his  New  Mathematical  Dictionary,  London,  1743,  speaks 
of  the  various  early  changes  of  the  calendar  and  then  says,   "  Pope 
Gregory  XIII.  pretended  to  reform  it  again,  and  ordered  his  account 
to  be  current,  as  it  still  is  in  all  the  Roman  Catholick  countries."     Much 
prejudice,  no  doubt,  lay  beneath  the  word  "pretended,"  and  the  word 
"  still,"  in  this  connection,  now  causes  a  smile. 

2  JOHN  KERSEY,  in  the  16th  Ed.  of  WINGATE'S  Arithmetick,  London, 
1735,  says  in  his  preface,  "For  the  Ease  and  Benefit  of  those  Learners 
that  desire  only  so  much  Skill  in  Arithmetick  as  is  useful  in  Accompts, 
Trade,  and  such  like  ordinary  Employments ;  the  Doctrine  of  Numbers 
(which,  in  the  First  Edition,  was  intermingled  with   Definitions  and 
Rules  concerning  Broken  Numbers,  commonly  called  Fractions)  is  now 
entirely  handled  a-part.  ...  So  that  now  Arithmetick  in  Whole  Numbers 


202  A   HISTORY    OF   MATHEMATICS 

of  arithmetics,  Wingate  for  instance,  show  how  to  find  the 
L.C.D.  in  the  addition  of  fractions,  the  majority  of  books 
take  for  the  C.D.  the  product  of  the  denominators.  Thus, 
Cocker  gives  8000  as  the  C.D.  of  f ,  £,  -f$,  %% ;  Dilworth  gives 

i  +  l  =  1T665  Hatton  gives  A +  *  =  «  =  «• 

It  was  the  universal  custom   to  treat   the   rule   of   three 

under  two  distinct  heads,  "  Eule  of  Three  Direct "  and  "  Kule 
of  Three  Inverse.'7  The  former  embraces  problems  like  this, 
"If  4  Students  spend  19  Pounds,  how  many  Pounds  will  8 
students  spend  at  the  same  Rate  of  Expence  ? "  ( Wingate.) 
The  inverse  rule  treats  questions  of  this  sort,  "If  8  horses 
will  be  maintained  12  Days  with  a  certain  Quantity  of  Prov- 
ender, How  many  Days  will  the  same  Quantity  maintain 
16  Horses?"  (Wingate.)  In  problems  like  the  former  the 
correct  answer  could  be  gotten  by  taking  the  three  numbers  in 
order  as  the  first  three  terms  of  a  proportion.  In  Wingate's 

Students       Pounds       Students 

notation,  we  would  have,  If  4  19  8.       But  in 

the  second  example  "  you  cannot  say  here  in  a  direct  propor- 
tion (as  before  in  the  Eule  of  Three  Direct)  as  8  to  16,  so  is 
12  to  another  Number  which  ought  to  be  in  that  Case  as  great 
again  as  12 ;  but  contrariwise  by  an  inverted  Proportion,  begin- 
ning with  the  last  Term  first ;  as  16  is  to  8,  so  is  12  to  another 
Number."  (Wingate,  1735,  p.  57.)  This  brings  out  very 
clearly  the  appropriateness  of  the  term  "Eule  of  Three 
inverse"  As  all  problems  were  classified  by  authors  under 
two  distinct  heads,  the  direct  rule  and  the  inverse  rule,  the 
pupil  could  get  correct  answers  by  a  purely  mechanical  pro- 
cess, without  being  worried  by  the  question,  under  which  rule 

is  plainly  and  fully  handled  before  any  Erit'rance  be  made  into  the 
craggy  Paths  of  Fractions,  at  the  Sight  of  which  some  Learners  are  so 
discouraged,  that  they  make  a  stand,  and  cry  out,  non  plus  ultra,  There's 
no  Progress  farther." 


MODERN   TIMES  203 

does  the  problem  come  ?  Thus  that  part  of  the  subject  which 
taxes  to  the  utmost  the  skill  of  the  modern  teacher  of  propor- 
tion, was  formerly  disposed  of  in  an  easy  manner.  No  heed 
was  paid  to  mental  discipline.  Nor  did  authors  care  what  the 
pupil  would  do  with  a  problem,  when  he  was  not  told  before- 
hand to  what  rule  it  belonged. 

Beginning  with  the  time  of  Cocker,  all  demonstrations  are 
carefully  omitted.  The  only  proofs  known  to  Dilworth  are  of 
this  kind,  "Multiplication  and  division  prove  each  other." 
The  only  evidence  we  could  find  that  John  Hill  was  aware 
of  the  existence  of  such  things  as  mathematical  demonstrations 
lies  in  the  following  passage,  "  So  ^  multiplied  by  ^  becomes  ^. 
See  this  demonstrated  in  Mr.  Leyburn's  Cursus  Mathematicus, 
page  38."  What  a  contrast  between  this  and  our  quotation  on 
fractions  from  Tonstall !  The  practice  of  referring  to  other 
works  was  more  common  in  arithmetics  then  than.  now. 
Cocker  refers  to  Kersey's  Appendix,  to  Wingate's  Arithmetic, 
to  Pitiscus's  Trigonometria,  and  to  Oughtred's  Clavis  for  the 
proof  of  the  rule  of  Double  Position.  Cocker  repeatedly  gives 
on  the  margin  Latin  quotations  from  the  Clavis,  from  Alsted's 
Mathematics,  or  Gemma-Frisius's  Methodus  Facilis  (Witten- 
berg, 1548). 

Toward  the  end  of  the  eighteenth  century  demonstrations 
begin  to  appear,  in  the  better  books,  but  they  are  often  placed 
at  the  foot  of  the  page,  beneath  a  horizontal  line,  the  rules  and 
examples  being  above  the  line.  By  this  arrangement  the 
author's  conscience  was  appeased,  while  the  teacher  and  pupil 
who  did  not  care  for  proofs  were  least  annoyed  by  their 
presence  and  could  easily  skip  them.  Moreover,  the  proofs 
and  explanations  were  not  adapted  to  the  young  mind ;  there 
was  no  trace  of  object-teaching;  the  presentation  was  too 
abstract.  True  reform,  both  in  England  and  America,  began 
only  with  the  introduction  of  Pestalozzian  ideas. 


204  A   HISTORY   OF   MATHEMATICS 

Mental  arithmetic  received  no  attention  in  England  before 
the  present  century,  but  in  Germany  it  was  introduced  during 
the  second  half  of  the  eighteenth  century.1 

Causes  which  Checked  the  Growth  of  Demonstrative  Arithmetic 
in  England 

Before  the  Reformation  there  was  little  or  nothing  accom- 
plished in  the  way  of  public  education  in  England.  In  the 
monasteries  some  instruction  was  given  by  monks,  but  we 
have  no  evidence  that  any  branch  of  mathematics  was  taught 
to  the  youth.2  In  1393  was  established  the  celebrated  "  public 
school,"  named  Winchester,  and  in  1440,  Eton.  In  the  six- 
teenth century,  on  the  suppression  of  the  monasteries, 
schools  were  founded  in  considerable  numbers  —  The  Mer- 
chant Taylors9  School,  Christ  Hospital,  Rugby,  Harrow,  and 
others  —  which  in  England  monopolize  the  title  "public 
schools"  and  for  centuries  have  served  for  the  education 
of  the  sons  mainly  of  the  nobility  and  gentry.3  In  these 
schools  the  ancient  classics  were  the  almost  exclusive  subjects 
of  study ;  mathematical  teaching  was  unknown  there.  Per- 
haps the  demands  of  every-day  life  forced  upon  the  boys  a 
knowledge  of  counting  and  of  the  very  simplest  computations, 
but  we  are  safe  in  saying  that,  before  the  close  of  the  last 
century,  the  ordinary  boy  of  England's  famous  public  schools 

1  UNGER,  p.  168. 

2  Some  idea  of  the  state  of  arithmetical  knowledge  may  be  gathered 
from  an  ancient  custom  at  Shrewsbury,  where  a  person  was  deemed  of 
age  when  he  knew  how  to  count  up  to  twelve  pence.     (Year-Books 
Edw.  L,  XX. -I.  Ed.  HORWOOD,  p.  220).     See  TYLOR'S  Primitive  Culture, 
New  York,  1889,  Vol.  I.,  p.  242. 

8  Consult  ISAAC  SHARPLESS,  English  Education,  New  York,  1892.; 
H.  F.  REDDALL,  School-boy  Life  in  Merrie  England,  New  York,  1891  ; 
JOHN  TIMES,  School-Days  of  Eminent  Men,  London. 


MODERN    TIMES  205 

could  not  divide  2021  by  43,  though  such  problems  had  been 
performed  centuries  before  according  to  the  teaching  of 
Brahmagupta  and  Bhaskara  by  boys  brought  up  on  the  far-off 
banks  of  the  Ganges.  It  has  been  reported  that  Charles  XII. 
of  Sweden  considered  a  man  ignorant  of  arithmetic  but  half 
a  man.  Such  was  not  the  sentiment  among  English  gentle- 
men. Not  only  was  arithmetic  unstudied  by  them,  but  con- 
sidered beneath  their  notice.  If  we  are  safe  in  following 
Thnbs's  account  of  a  book  of  1622,  entitled  Peacham's  Compleat 
Gentleman,  which  enumerates  subjects  at  that  time  among 
the  becoming  accomplishments  of  an  English  man  of  rank, 
then  it  appears  that  the  elements  of  astronomy,  geometry,  and 
mechanics  were  studies  beginning  to  demand  a  gentleman's 
attention,  while  arithmetic  still  remained  untouched.1  Listen 
to  another  writer,  Edmund  Wells.  In  his  Young  Gentleman's 
Course  in  Mathematicks,  London,  1714,  this  able  author  aims 
to  provide  for  gentlemanly  education  as  opposed  to  that  of 
"  the  meaner  part  of  mankind." 2  He  expects  those  whom  God 
has  relieved  from  the  necessity  of  working,  to  exercise  their 
faculties  to  his  greater  glory.  But  they  must  not  "  be  so  Brisk 
and  Airy,  as  to  think,  that  the  knowing  how  to  cast  Accompt 
is  requisite  only  for  such  Underlings  as  Shop-keepers  or  Trades- 
men," and  if  only  for  the  sake  of  taking  care  of  themselves,  "  no 
gentleman  ought  to  think  Arithmetick  below  Him  that  do's 
not  think  an  Estate  below  Him."  All  the  information  we 
could  find  respecting  the  education  of  the  upper  classes  points 
to  the  conclusion  that  arithmetic  was  neglected,  and  that  De 
Morgan3  was  right  in  his  statement  that  as  late  as  the  eigh- 
teenth century  there  could  have  been  no  such  thing  as  a  teacher 
of  arithmetic  in  schools  like  Eton.  In  1750,  Warren  Hastings, 

1  TIMES,  School-Days,  p.  101. 

2  DE  MORGAN,  AritU.  Books,  p.  64. 
8  Arith.  Books,  p.  76. 


206  A  HISTORY  Otf   MATHEMATICS 

who  had  been  attending  Westminster,  was  put  into  a  commer- 
cial school,  that  he  might  study  arithmetic  and  book-keeping 
before  sailing  for  Bengal. 

At  the  universities  little  was  done  in  mathematics  before 
the  middle  of  the  seventeenth  century.  It  would  seem  that 
Tonstall's  work  was  used  at  Cambridge  about  1550,  but  in 
1570,  during  the  reign  of  Queen  Elizabeth,  fresh  statutes  were 
given,  excluding  all  mathematics  from  the  course  of  undergradu- 
ates, presumably  because  this  study  pertained  to  practical  life, 
and  could,  therefore,  have  no  claim  to  attention  in  a  university.1 

The  commercial  element  in  the  arithmetics  and  algebras  of 
early  times  was  certainly  very  strong.  Observe  how  the  Arab 
Muhammed  ibn  Musa  and  the  Italian  writers  discourse  on 
questions  of  money,  partnership,  and  legacies.  Significant 
is  the  fact  that  the  earliest  English  algebra  is  dedicated  by 
Eecorde  to  the  company  of  merchant  adventurers  trading  to 
Moscow.2  Wright,  in  his  English  edition  of  Napier's  loga- 
rithms, likewise  appeals  to  the  commercial  classes :  "  To  the 
Eight  Honourable  And  Eight  Worshipfvll  Company  Of  Mer- 
chants of  London  trading  to  the  East-Indies,  Samvel  Wright 
wisheth  all  prosperitie  in  this  life  and  happinesse  in  the  life  to 
come." 3  It  is  also  significant  that  the  first  headmaster  of  the 
Merchant  Taylors'  School,  a  reformer  with  views  in  advance  of 
his  age,  in  a  book  of  1581  speaks  of  mathematical  instruction, 
thinks  that  a  few  of  the  most  earnest  and  gifted  students  might 
hope  to  attain  a  knowledge  of  geometry  and  arithmetic  from 
Euclid's  Elements,  but  fails  to  notice  Eecorde's  arithmetic, 
which  since  1540  had  appeared  in  edition  after  edition.4  Clas- 


1  BALL,  Mathematics  at  Cambridge,  p.  13. 

2  G.  HEPPEL  in  19th  General  Report  (1893)  of  the  A.  I.  G.  T.,  pp. 
and  27. 

8  NAPIER'S  Construction  (MACDONALD'S  Ed.),  p.  145. 
*  G.  HEPPEL,  op.  cit.,  p.  28. 


MODERN   TIMES  207 

sical  men  were  evidently  not  in  touch  with  the  new  in  the 
mathematics  of  that  time. 

This  scorn  and  ignorance  of  the  art  of  computation  by  all 
but  commercial  classes  is  seen  in  Germany  as  well  as  England. 
Kastner1  speaks  of  it  in  connection  with  German  Latin 
schools;  linger  refers  to  it  repeatedly.2 

It  was  not  before  the  present  century  that  arithmetic  and 
other  branches  of  mathematics  found  admission  into  England's 
public  schools.  At  Harrow  "vulgar  fractions,  Euclid,  geog- 
raphy, and  modern  history  were  first  studied "  in  1829.3  At 
the  Merchant  Taylors'  School  "mathematics,  writing,  and 
arithmetic  were  added  in  1829." 4  At  Eton  "  mathematics  was 
not  compulsory  till  1851." 6  The  movement  against  the  ex- 
clusive classicism  of  the  schools  was  led  by  Dr.  Thomas 
Arnold  of  Rugby,  the  father  of  Matthew  Arnold.  Dr.  Arnold 
favoured  the  introduction  of  mathematics,  science,  history,  and 
politics. 

Since  the  art  of  calculation  was  no  more  considered  a  part  of 
a  liberal  education  than  was  the  art  of  shoe-making,  it  is 
natural  to  find  the  study  of  arithmetic  relegated  to  the  com- 
mercial schools.  The  poor  boy  sometimes  studied  it ;  the  rich 
boy  did  not  need  it.  In  Latin  schools  it  was  unknown,  but  in 
schools  for  the  poor  it  was  sometimes  taught;  for  example, 
in  a  "  free  grammar  school  founded  by  a  grocer  of  London  in 
1553  for  thirty  '  of  the  poorest  men's  sons '  of  Guilford,  to  be 
taught  to  read  and  write  English  and  cast  accounts  perfectly, 
so  that  they  should  be  fitted  for  apprentices." 6  That  a 
science,  ignored  as  a  mental  discipline,  and  studied  merely  as 
an  aid  to  material  gain,  should  fail  to  receive  fuller  develop- 
ment, is  not  strange.  It  was,  in  fact,  very  natural  that  it 

1  Geschichte,  III.,  p.  429.  4  TIMES,  p.  84. 

2  See  pp.  5,  24,  112,  140,  144.  5  SHARPLESS,  p.  144. 
8  REDDALL,  p.  228.                                  •  TIMES,  p.  83. 


208  A   HISTORY   OF   MATHEMATICS 

should  entirely  discard  those  features  belonging  more  prop 
erly  to  a  science  and  assume  the  form  of  an  art.  So  arithmetic 
reduced  itself  to  a  mere  collection  of  rules.  The  ancient 
Carthaginians,  like  the  English,  studied  arithmetic,  but  did 
not  develop  it  as  a  science.  "  It  is  a  beautiful  testimony  to 
the  quality  of  the  Greek  mind  that  Plato  and  others  assign  as 
a  cause  of  the  low  state  of  arithmetic  and  mathematics  among 
the  Phoenicians  .  .  .  the  want  of  free  and  disinterested  inves- 
tigation." * 

It  will  be  observed  that  during  the  period  under  considera- 
tion, the  best  English  mathematical  minds  did  not  make  their 
influence  felt  in  arithmetic.  The  best  intellects  held  aloof 
from  the  elementary  teacher;  arithmetical  texts  were  written 
by  men  of  limited  education.  During  the  seventeenth  and 
eighteenth  centuries  England  had  no  Tonstall  and  Eecorde, 
no  Stifel  and  Eegiomontanus,  no  Pacioli  and  Tartaglia,  to 
compose  her  arithmetical  books.  To  be  sure,  she  had  her 
Wallis,  her  Newton,  Cotes,  Hook,  Taylor,  Maclaurin,  De 
Moivre,  but  they  wrote  no  books  for  elementary  schools ;  their 
influence  on  arithmetical  teaching  was  naught.  Contrast  this 
period  with  the  present  century.  Think  of  the  pains  taken 
by  Augustus  De  Morgan  to  reform  elementary  mathematical 
instruction.  The  man  who  could  write  a  brilliant  work  on 
the  Calculus,  who  could  make  new  discoveries  in  advanced 
algebra,  series,  and  in  logic,  was  the  man  who  translated 
Bourdon's  arithmetic  from  the  French,  composed  an  arith- 
metic and  elementary  algebra  for  younger  students,  and 
endeavoured  to  simplify,  without  loss  of  rigour,  the  Euclidean 
geometry.  Again,  run  over  the  list  of  members  of  the  "  Asso- 
ciation for  the  Improvement  of  Geometrical  Teaching,"  and 
of  the  Committee  of  the  British  Association  on  geometrical 

1  J.  K.  F.  ROSENKRANZ,  Philosophy  of  Education,  translated  by  A.  C. 
BRACKETT,  New  York,  1888,  p.  215. 


MODERN   TIMES  209 

teaching,  and  you  will  find  in  it  England's  most  brilliant 
mathematicians  of  our  time. 

The  imperfect  interchange  of  ideas  between  writers  on 
advanced  and  those  on  elementary  subjects  is  exhibited  in 
the  mathematical  works  of  John  Ward  of  Chester.  He  pub- 
lished in  1695  a  Compendium  of  Algebra,  and  in  1706  his 
Young  Mathematician' 's  Guide.  The  latter  work  appeared  in 
the  12th  edition  in  1771,  was  widely  read  in  Great  Britain, 
and  well  approved  in  the  universities  of  England,  Scotland, 
and  Ireland.  It  was  once  a  favourite  text-book  in  Ameri- 
can colleges,  being  used  as  early  as  1737  at  Harvard  College, 
and  as  late  as  1787  at  Yale  and  Dartmouth.  In  475  pages, 
the  book  covered  the  subjects  of  Arithmetic,  Algebra,  Geom- 
etry, Conic  Sections,  and  Arithmetic  of  Infinites,  giving,  of 
course,  the  mere  rudiments  of  each. 

Ward  shows  how  to  raise  a  binomial  to  positive  integral 
powers  "  without  the  trouble  of  continued  involution "  and 
remarks  that  when  he  published  this  method  in  his  Compen- 
dium of  Algebra  he  thought  that  he  was  the  first  inventor  of 
it,  but  that  since  he  has  found  in  Wallis's  History  of  Algebra 
"that  the  learned  Sir  Isaac  Newton  had  discovered  it  long 
before."  It  took  a  quarter  of  a  century  for  the  news  of 
Newton's  binomial  discovery  to  reach  John  Ward.  More- 
over, it  looks  very  odd  to  see  in  Ward's  Guide  of  1771  —  over 
a  century  after  Newton's  discovery  of  fluxions  —  a  sort  of 
integral  calculus,  such  as  was  employed  by  Wallis,  Cavalieri, 
Fermat,  and  Roberval  before  the  invention  of  fluxions. 

It  is  difficult  to  discover  a  time  when  in  any  civilized  country 
advanced  and  elementary  writers  on  mathematics  were  more 
thoroughly  out  of  touch  with  each  other  than  in  England  dur- 
ing the  two  centuries  and  a  half  preceding  1800.  We  can 
think  of  no  other  instance  in  science  in  which  the  failure  of 
the  best  minds  to  influence  the  average  has  led  to  such 

F 


210  A  HISTORY   OF  MATHEMATICS 

lamentable  results.  In  recent  times  we  have  heard  it  deplored 
that  in  some  countries  practical  chemistry  does  not  avail  itself 
of  the  results  of  theoretical  chemistry.  Fears  have  been 
expressed  of  a  coming  schism  between  applied  higher  mathe- 
matics and  theoretical  higher  mathematics.  But  thus  far 
these  evils  appear  insignificant,  as  compared  with  the  wide- 
spread repression  and  destruction  of  demonstrative  arithmetic, 
arising  from  the  failure  of  higher  minds  to  guide  the  rank 
and  file.  Neglected  by  the  great  thinkers  of  the  day,  scorned 
by  the  people  of  rank,  urged  onward  by  considerations  of 
purely  material  gain,  arithmetical  writers  in  England  (as 
also  in  Germany  and  France)  were  led  into  a  course  which 
for  centuries  blotted  the  pages  of  educational  history. 

In  those  days  English  and  German  boys  often  prepared 
for  a  business  career  by  attending  schools  for  writing  and 
arithmetic.  During  the  Middle  Ages  and  also  long  after  the 
invention  of  printing,  the  art  of  writing  was  held  in  high 
esteem.  In  writing  schools  much  attention  was  given  to 
fancy  writing.  The  schools  embracing  both  subjects  are 
always  named  schools  for  "writing  and  arithmetic,"  never 
"  arithmetic  and  writing."  The  teacher  was  called  "  writing- 
master  and  arithmetician."  Cocker  was  skilful  with  the  pen, 
and  wrote  many  more  books  on  calligraphy  than  on  arithmetic. 
As  to  the  quality  of  the  teachers,  Peacham  in  his  Compleat 
Gentleman  (1622)  stigmatizes  the  schoolmasters  in  his  own 
day  as  rough  and  even  barbarous  to  their  pupils.  Domestic 
tutors  he  represents  as  still  worse.1  The  following  from 
an  arithmetic  of  William  Webster,  London,  1740,  gives  an 
idea  of  the  origin  of  many  commercial  schools.2  "When  a 
Man  has  tried  all  Shifts,  and  still  failed,  if  he  can  but  scratch 
out  anything  like  a  fair  Character,  tho  never  so  stiff  and 

1  TIMES,  p.  101  2  DE  MORGAN,  Arith.  Books,  p.  t>9. 


MODERN  TIMES  211 

unnatural  and  has  got  but  Arithmetick  enough  in  his  Head  to 
compute  the  Minutes  in  a  Year,  or  the  Inches  in  a  Mile,  he 
makes  his  last  Recourse  to  a  Garret,  and  with  the  Painter's 
Help,  sets  up  for  a  Teacher  of  Writing  and  Arithmetick;  where, 
by  the  Bait  of  low  Prices,  he  perhaps  gathers  a  Number  of 
Scholars."  No  doubt  in  previous  centuries,  as  in  all  times5 
there  were  some  good  teachers,  but  the  large  mass  of  school- 
masters in  England,  Germany,  and  the  American  colonies  were 
of  the  type  described  in  the  above  extract.  Some  of  the  more 
ambitious  and  successful  of  these  teachers  wrote  the  arithmetics 
for  the  schools.  No  wonder  that  arithmetical  authorship  and 
teaching  were  at  a  low  ebb. 

To  summarize,  the  causes  which  checked  the  growth  of 
demonstrative  arithmetic  are  as  follows : 

(1)  Arithmetic  was  not  studied  for  its  own  sake,  nor  valued 
for  the  mental  discipline  which  it  affords,  and  was,  consequently, 
learned  only  by  the  commercial  classes,  because  of  the  material 
gain  derived  from  a  knowledge  of  arithmetical  rules. 

(2)  The  best  minds  failed  to  influence  and  guide  the  average 
minds  in  arithmetical  authorship. 

Reforms  in  Arithmetical  Teaching 

A  great  reform  in  elementary  education  was  initiated  in 
Switzerland  and  Germany  at  the  beginnning  of  the  nine- 
teenth century.  Pestalozzi  emphasized  in  all  instruction  the 
necessity  of  object-teaching.  "  Of  the  instruction  at  Yverdun, 
the  most  successful  in  the  opinion  of  those  who  visited  the 
school,  was  the  instruction  in  arithmetic.  The  children  are 
described  as  performing  with  great  rapidity  very  difficult  tasks 
in  head-calculation.  Pestalozzi  based  his  method  here,  as  in 
other  subjects,  on  the  principle  that  the  individual  should  be 
brought  to  the  knowledge  by  the  road  similar  to  that  which 


212  A   HISTORY   OP   MATHEMATICS 

the  whole  race  had  used  in  founding  the  science.  Actual 
counting  of  things  preceded  the  first  Cocker,  as  actual  meas- 
urement of  land  preceded  the  original  Euclid.  The  child 
must  be  taught  to  count  things  and  to  find  out  the  various 
processes  experimentally  in  the  concrete,  before  he  is  given 
any  abstract  rule,  or  is  put  to  abstract  exercises.  This  plan 
is  now  adopted  in  German  schools,  and  many  ingenious  con- 
trivances have  been  introduced  by  which  the  combinations  of 
things  can  be  presented  to  the  children's  sight." *  Much 
remained  to  be  done  by  the  followers  of  Pestalozzi  in  the  way 
of  practical  realization  of  his  ideas.  Moreover,  a  readjust- 
ment of  the  course  of  instruction  was  needed,  for  Pestalozzi 
quite  ignored  those  parts  of  arithmetic  which  are  applied  in 
every-day  life.  Until  about  1840  Pestalozzi 's  notions  were 
followed  more  or  less  closely.  In  1842  appeared  A.  W.  Grube's 
Leitfaden,  which  was  based  on  Pestalozzi's  idea  of  object- 
teaching,  but,  instead  of  taking  up  addition,  subtraction,  mul- 
tiplication, and  division  in  the  order  as  here  given,  advocated 
the  exclusion  of  the  larger  numbers  at  the  start,  and  the  teach- 
ing of  all  four  processes  in  connection  with  the  first  circle 
of  numbers  (say,  the  numbers  1  to  10)  before  proceeding  to 
a  larger  circle.  For  a  time  Grube's  method  was  tried  in 
Germany,  but  it  soon  met  with  determined  opposition.  The 
experience  of  both  teachers  and  students  seems  to  be  that,  to 
reach  satisfactory  results,  an  extraordinary  expenditure  of 
energy  is  demanded.  In  the  language  of  physics,  Grube's 
method  seems  to  be  an  engine  having  a  low  efficiency.2 

Into  conservative  England  Pestalozzian  ideas  found  tardy  ad- 

1  R.  H.  QUICK,  Educational  Reformers,  1879,  p.  191. 

2  For  the  history  of  arithmetical  teaching  in   Germany  during  the 
nineteenth  century,  see  UNGER,  pp.  175-233.     For  a  discussion  of  the 
psychological  bearing  of  Grube's  method,  see  MCLELLAN  AND  DEWEY, 
Psychology  of  Number,  1895,  p.  80  et  seq. 


MODERN   TIMES  213 

mittance.  At  the  time  when  De  Morgan  began  to  write  (about 
1830)  arithmetical  teaching  had  not  risen  far  above  the  level 
of  the  eighteenth  century.  In  more  recent  time  the  teach- 
ing of  arithmetic  has  been  a  subject  of  discussion  by  the 
"  Association  for  the  Improvement  of  Geometrical  Teaching." J 
Among  the  subjects  under  consideration  have  been  the  mul- 
tiplication and  division  of  concrete  quantities,  the  approxi- 
mate multiplication  of  decimals  (by  leaving  off  the  tail  of  the 
product),  and  modification  in  the  processes  of  subtraction,  mul- 
tiplication, and  division.  The  new  modes  of  subtraction  and 
division  are  called  in  Germany  the  "  Austrian  methods,"  be- 
cause the  Austrians  were  the  first  to  adopt  them.  In  England 
this  mode  of  division  goes  by  the  name  of  "  Italian  method." 2 
The  "Austrian "  method  of  subtraction  is  simply  this :  It 
shall  be  performed  in  the  same  way  as  "  change  "  is  given  in 
a  store  by  adding  from  the  lower  to  the  higher  instead  of 
passing  from  the  higher  to  the  lower  by  mental  subtraction. 
Thus,  in  76  —  49,  say  "  nine  and  seven  are  sixteen,  five  and 
two  are  seven."  The  practice  of  adding  1  to  the  4,  instead 
of  subtracting  1  from  7,  was  quite  common  during  the  Renais- 
sance. It  was  used,  for  instance,  by  Maximus  Planudes 
Georg  Peuerbach,  and  Adam  Eiese.  In  Adam  Eiese's  works 
there  is  also  an  approach  to  the  determination  of  the  differ- 
ence by  figuring  from  the  subtrahend  upwards.  In  the  above 
example,  he  would  subtract  9  from  10  and  add  the  remainder 
1  to  the  minuend  6,  thus  obtaining  the  answer  7.3  A  recent 
American  text-book  on  algebra  explains  subtraction  on  the 
principle  of  the  "  Austrian  method." 

1  See  the  General  Reports  since  1888,  especially  those  for  1892  and  1893. 

2  The  name  "  Italian  method  "  was  traced  by  Mr.  Langley  back  to  an 
English  arithmetic  of  1730.    See  General  Eeport  of  A.  I.  G.  T. ,  1892,  p.  34. 

3  ARNO  SADOWSKT,  Die  osterreichische  Rechenmethode  in  padagogischer 
und  historischer  Belcuchtung,  Konigsberg,  1892,  p.  13. 


214  A  HISTORY  OF  MATHEMATICS 

In  multiplication,  the  recommendation  is  to  begin,  as  in  alge- 
bra, with  the  figure  of  highest  denomination  in  the  multiplier. 
The  great  advantage  of  this  shows  itself  in  decimal  multiplica- 
tion in  case  we  desire  only  an  approximate  answer  correct  to, 
say,  three  or  five  decimal  places.  We  have  seen  that  this  method 
was  much  used  by  the  Florentines  and  was  called  by  Pacioli 
"by  the  little  castle."  Of  the  various  multiplication  processes 
of  the  Renaissance,  the  fittest  failed  to  survive.1  The  process 
now  advocated  was  taught  by  Nicholas  Pike2  in  the  following 
example :  "  It  is  required  to  multiply  56.7534916  by  5.376928, 
and  to  retain  only  five  places  of  decimals  in  the  product." 

The  "Austrian"  or  "Italian"  method 
56.7534916 
82  9673  5  °*  division  simply  calls  for  the  multiph- 

28376746  cation  of  the  divisor  and  the  subtraction 

1702605  .  .  .  from   the   dividend   simultaneously,  so 

397274  ....  that    only    the    remainder    is    written 

down  on  paper.     See  our   illustration. 

•Q3  It  is  doubtful  whether  this  method  is 

45 preferable  to  our  old  method,  except 


305.15943  for    naturally    rapid    computers.      We 

978)272862(279  fear  that  th6  Sl°W  COmPutor  saves  PaPer 

by  it,  at  the  expense  of  mental  energy. 


8802  The  "  Austrian  method  of  division  "   is 

1  This  process  was  favoured  by  Lagrange.     He  says  :  "But  nothing 
compels  us  to  begin  with  the  right  side  of  the  multiplier  ;  we  may  as  well 
begin  with  the  left  side,  and  I  truly  fail  to  see  why  this  method  is  not 
preferred,  for  it  has  the  great  advantage  of  giving  the  places  of  highest 
value  first  ;  in  the  multiplication  of  large  numbers  we  are  often  interested 
most  in  the  highest  places."     See  H.  NIEDERMULLER,  Lagrange  's  Mathe- 
matische   Elementarvorlesungen.       Deutsche    Separatausgabe,    Leipzig, 
1880,  p.  23.     This  publication,  giving  five  lectures  on  arithmetic  and 
algebra,  delivered  by  the  great  Lagrange  at  the  Normal  School  in  Paris, 
in  1795,  is  very  interesting  for  several  reasons. 

2  N.  PIKE,  Arithmetic,  abridged  for  the  use  of  schools,  3d  Ed.,  Worces- 
ter, 1798,  p.  92. 


MODERN   TIMES  215 

not  entirely  new ;  in  the  "  galley  "  or  "  scratch "  method,  the 
partial  products  were  not  written  down,  but  at  once  subtracted, 
and  only  the  remainder  noted.  In  principle,  the  "Austrian 
method"  was  practised  by  the  Hindus,  of  whose  process  the 
"  scratch  "  method  is  a  graphical  representation.1 

Arithmetic  in  the  United  States 

Weights  and  Measures. — The  weights  and  measures  intro- 
duced into  the  United  States  were  derived  from  the  English. 
"It  was  from  the  standards  of  the  exchequer  that  all  the 
weights  and  measures  of  the  United  States  were  derived,  until 
Congress  fixed  the  standard.  Louisiana  at  first  recognized 
standards  derived  from  the  French,  but  in  1814  the  United 
States  revenue  standards  were  established  by  law."2  The 
actual  standards  used  in  the  several  states  and  in  the  custom- 
houses were,  however,  found  to  be  very  inaccurate.  In  the 
construction  of  accurate  standards  for  American  use,  our  Gov- 
ernment engaged  the  services  of  Ferdinand  Rudolph  Hassler, 
a  Swiss  by  birth  and  training,  and  a  skilful  experimental- 
ist.3 The  work  of  actual  construction  was  begun  in  1835.4 
In  1836,  carefully  constructed  standards  were  distributed  to 
the  custom-houses  and  furnished  the  means  of  uniformity 
in  the  collection  of  the  customs.  Moreover,  accurate  stand- 
ards were  distributed  by  the  general  Government  to  the 

1  For  further  information  regarding  the  "Austrian"  method  of  sub- 
traction and  division,  see  SADOWSKI,  op.  cit.  ;  UNGER,  pp.  213-218. 

2  Report  of  the  Secretary  of  the  Treasury  on  the  Construction  and  Dis- 
tribution of  Weights  and  Measures.     Ex.  Doc.  No.  27,  1857,  p.  36. 

8  Consult  the  translation  from  the  German  of  Memoirs  of  Ferdinand 
Rudolph  Hassler,  by  EMIL  ZSCHOKKE,  published  in  Aarau,  Switzerland, 
1877  ;  with  supplementary  documents,  published  at  Nice,  1882.  See  also 
Teach,  and  Hist,  of  Math,  in  the  U.  S.,  pp.  286-289. 

4  Ex.  Doc.  No.  84  (Report  by  A.  D.  Bache  for  1846-47),  p.  2. 


216  A   HISTORY   OF   MATHEMATICS 

various  States,  with  the  view  of  securing  greater  uniformity. 
It  was  recommended  that  each  State  prepare  for  them  a 
fire-proof  room  and  place  them  "under  the  charge  of  some 
scientific  person  who  would  attend  to  their  use  and  safe- 
keeping." 

In  1866  Congress  authorized  the  use  of  the  Metric  System 
in  the  United  States,  but  unfortunately  stopped  here  and 
allowed  the  nations  of  Continental  Europe  to  advance  far 
beyond  us  by  their  adoption  of  the  Metric  System  to  the  exclu- 
sion of  all  older  systems. 

In  the  currency  of  the  American  colonies  there  existed  great 
diversity  and  confusion.  "  At  the  time  of  the  adoption  of  our 
decimal  currency  by  Congress,  in  1786,  the  colonial  currency  or 
bills  of  credit,  issued  by  the  colonies,  had  depreciated  in  value, 
and  this  depreciation,  being  unequal  in  the  different  colonies, 
gave  rise  to  the  different  values  of  the  State  currencies."1 
Inasmuch  as  all  our  early  arithmetics  were  "  practical "  arith- 
metics, they,  of  course,  gave  rules  for  the  "  reduction  of  coin." 
Thus  Pike's  Arithmetic2  devotes  twenty-two  pages  to  the 
statement  and  illustration  of  rules  for  reducing  "Newhamp- 
shire,  Massachusetts,  Rhodeisland,  Connecticut,  and  Virginia 
currency"  (1)  to  "Federal  Money,"  (2)  to  "Newyork  and 
Northcarolina  currency,"  (3)  to  "Pennsylvania,  Newjersey, 
Delaware,  and  Maryland  currency,  .  .  ."  (6)  to  "  Irish  money," 
(7)  to  "  Canada  and  Novascotia  currency,"  (8)  to  "  Livres  Tour- 
nois,"  (9)  to  "  Spanish  milled  dollars."  Then  follow  rules  for 
reducing  Federal  Money  to  "Newengland  and  Virginia  cur- 
rency," etc.  It  is  easy  to  see  how  a  large  share  of  the  pupil's 
time  was  absorbed  in  the  mastery  of  these  rules.  The  chap- 
ters on  reduction  of  coins,  on  duodecimals,  alligation,  etc., 

1  ROBINSON,  Progressive  Higher  Arithmetic,  1874,  p.  190. 

2  The  New  and  Complete  System  of  Arithmetic,  abridged  for  the  use 
of  schools,  3d  Ed.  1798,  Worcester,  pp.  117-139. 


MODERN   TIMES  217 

give  evidence  of  the  homage  that  Education  was  forced  to 
pay  to  Practical  Life,  at  the  sacrifice  of  matter  better  fitted 
to  develop  the  mind  of  youth.  With  a  view  of  supplying 
the  information  needed  by  merchants  in  business,  arithmetics 
discussed  such  subjects  as  the  United  States  Securities,  the 
various  rules  adopted  by  the  United  States,  and  by  the 
State  governments  on  partial  payments. 

Authors  and  Books.  —  The  first  arithmetics  used  in  the 
American  colonies  were  English  works :  Cocker,  Hodder, 
Dilworth,  "George  Fisher"  (Mrs.  Slack),  Daniel  Fenning.1 
The  earliest  arithmetic  written  and  printed  in  America  ap- 
peared anonymously  in  Boston  in  1729.  Though  a  work  of 
considerable  merit,  it  seems  to  have  been  used  very  little ;  in 
early  records  we  have  found  no  reference  to  it ;  fifty  years 
later,  at  the  publication  of  Pike's  Arithmetic,  the  former  work 
was  completely  forgotten,  and  Pike's  was  declared  to  be  the 
earliest  American  arithmetic.  Of  the  1729  publication  there 
are  two  copies  in  the  Harvard  Library  and  one  in  the  Congres- 
sional Library.2  In  Appleton's  Cyclopaedia  of  American  Biog- 
raphy its  authorship  is  ascribed  without  reserve  to  Isaac  Green- 
wood, then  professor  of  mathematics  at  Harvard  College,  but 
on  the  title-page  of  one  of  the  copies  in  the  Harvard  Library, 
is  written  the  following:  "Supposed  that  Sam!  Greenwood 
was  the  author  thereof,  by  others  said  to  be  by  Isaac  Green- 
wood." In  1788  appeared  at  Newburyport  the  New  and 
Complete  System  of  Arithmetic  by  Nicholas  Pike  (1743-1819), 
a  graduate  of  Harvard  College.3  It  was  intended  for  advanced 
schools,  and  contained,  besides  the  ordinary  subjects  of  that 
time,  logarithms,  trigonometry,  algebra,  and  conic  sections; 
but  these  latter  subjects  were  so  briefly  treated  as  to  possess 
little  value.  In  the  "Abridgment  for  the  Use  of  Schools," 

1  See  Teach,  and  Hist,  of  Math,  in  the  U.S.,  pp.  12-16. 

2  Ibidem,  p.  14.  3  Ibidem,  pp.  45,  46. 


218  A    HISTORY   OF   MATHEMATICS 

which  was  brought  out  at  Worcester  in  1793,  the  larger  work 
is  spoken  of  in  the  preface  as  "  now  used  as  a  classical  book  in 
all  the  Newengland  Universities."  A  recent  writer1  makes 
Pike  responsible  for  all  the  abuses  in  arithmetical  teaching 
that  prevailed  in  early  American  schools.  To  us  this  con- 
demnation of  Pike  seems  wholly  unjust.  It  is  unmerited, 
even  if  we  admit  that  Pike  was  in  no  sense  a  reformer  among 
arithmetical  authors.  Most  of  the  evils  in  question  have  a  far 
remoter  origin  than  the  time  of  Pike.  Our  author  is  fully  up 
to  the  standard  of  English  works  of  that  date.  He  can  110 
more  be  blamed  by  us  for  giving  the  aliquot  parts  of  pounds 
and  shillings,  for  stating  rules  for  "  tare  and  trett,"  for  dis- 
cussing the  "  reduction  of  coins,"  than  the  future  historian  can 
blame  works  of  the  present  time  for  treating  of  such  atrocious 
relations  as  that  3  ft.  =  1  yd.,  51  yds.  =  1  rd.,  30£  sq.  yds.  =  1 
sq.  rd.,  etc.  So  long  as  this  free  and  independent  people 
chooses  to  be  tied  down  to  such  relics  of  barbarism,  the  arith- 
matician  cannot  do  otherwise  than  supply  the  means  of  acquir- 
ing the  precious  knowledge. 

At  the  beginning  of  the  nineteenth  century  there  were  three 
"  great  arithmeticians  "  in  the  United  States :  Nicholas  Pike, 
Daniel  Adams,  and  Nathan  Daboll.  The  arithmetics  of  Adams 
(1801)  and  of  Daboll  (1800)  paid  more  attention  than  that  of 
Pike  to  Federal  Money.  Peter  Parley  tells  us  that  in  conse- 
quence of  the  general  use,  for  over  a  century,  of  Dilworth  in 
American  schools,  pounds,  shillings,  and  pence  were  classical, 
and  dollars  and  cents  vulgar  for  several  succeeding  generations. 
"  I  would  not  give  a  penny  for  it "  was  genteel ;  "  I  would  not 
give  a  cent  for  it "  was  plebeian. 

Reform  in  arithmetical  teaching  in  the  United  States  did 
not  begin  until  the  publication  by  Warren  Colburn,  in  1821, 

1  GEORGE  H.  MARTIN,  The  Evolution  of  the  Massachusetts  Public 
School  System,  p.  102. 


MODERN   TIMES  219 

of  the  Intellectual  Arithmetic.1  This  was  the  first  fruit  of 
Pestalozzian  ideas  on  American  soil.  Like  Pestalozzi,  Col- 
burn's  great  success  lay  in  the  treatment  of  mental  arithmetic. 
The  success  of  this  little  book  was  extraordinary.  But  Ameri- 
can teachers  in  Colburn's  time,  and  long  after,  never  quite 
succeeded  in  successfully  engrafting  Pestalozzian  principles 
on  written  arithmetic.  Too  much  time  was  assigned  to  arith- 
metic in  schools.  There  was  too  little  object-teaching;  either 
too  much  abstruse  reasoning,2  or  no  reasoning  at  all ;  too  little 
attention  to  the  art  of  rapid  and  accurate  computation;  too 
much  attention  to  the  technicalities  of  commercial  arithmetic. 
During  the  last  ten  years,  however,  desirable  reforms  have 
been  introduced.3 

"Pleasant  and  Diverting  Questions" 

In  English  and  American  editions  of  Dilworth,  as  also  in 
Daniel  Adams's  Scholar's  Arithmetic*  we  find  a  curious  col- 
lection of  "  Pleasant  and  diverting  questions."  We  have  all 
heard  of  the  farmer,  who,  having  a  fox,  a  goose,  and  a  peck 
of  corn,  wished  to  cross  a  river;  but,  being  able  to  carry 
only  one  at  a  time,  was  confounded  as  to  how  he  should  take 

1  "  WARREN  COLBURN'S  First  Lessons  have  been  abused  by  being  put 
in  the  hands  of  children  too  early,  and  has  been  productive  of  almost  as 
much  harm  as  good."  —  REV.  THOMAS  HILL,  The  True  Order  of  Studies, 
1876,  p.  42. 

2  "The  teacher  who  has  been  accustomed  to  the  modern  erroneous 
method  of  teaching  a  child  to  reason  out  his  processes  from  the  beginning 
may  be  assured  this  method  of  gaining  facility  in  the  operations,  before 
attempting  to  explain  them,  is  the  method  of  Nature  ;  and  that  it  is  not 
only  much  pleasanter  to  the  child,  but  that  it  will  make  a  better  mathe- 
matician of  him."  —  T.  HILL,  op.  cit.,  p.  45. 

8  For  a  more  detailed  history  of  arithmetical  teaching,  see  Teach,  and 
Hist,  of  Math,  in  the  U.  S. 

*  Seventh  Ed.,  Montpelier,  Vt.,  1812,  p.  210. 


220  A   HISTORY   OF   MATHEMATICS 

them  across  so  that  the  fox  should  not  devour  the  goose,  nor 
the  goose  the  corn.  Who  has  not  been  entertained  by  the 
problem,  how  three  jealous  husbands  with  their  wives  may 
cross  a  river  in  a  boat  holding  only  two,  so  that  none  of  the 
three  wives  shall  be  found  in  company  of  one  or  two  men 
unless  her  husband  be  present  ?  Who  has  not  attempted  to 
place  nine  digits  in  a  square  so  that  any  three  figures  in  a 
line  may  make  just  15?  None  of  us,  perhaps,  at  first  sus- 
pected the  great  antiquity  of  these  apparently  new-born 
creatures  of  fancy.  Some  of  these  puzzles  are  taken  by 
Dilworth  from  Kersey's  edition  of  Wingate.  Kersey  refers 
the  reader  to  "  the  most  ingenious  "  Gaspar  Backet  de  Meziriac 
in  his  little  book,  Problemes  plaisants  et  delectables  qui  se  font 
par  les  nombres  (Lyons,  1624),  which  is  still  largely  read. 
The  first  of  the  above  puzzles  was  probably  known  to  Charle- 
magne, for  it  appears  in  Alcuin's  (?)  Propositiones  ad  acuendos 
juvenes,  in  the  modified  version  of  the  wolf,  goat,  and  cabbage 
puzzle.  The  three  jealous  husbands  and  their  wives  were 
known  to  Tartaglia,  who  also  proposes  the  same  question  with 
four  husbands  and  four  wives.1  We  take  these  to  be  modified 
and  improved  versions  of  the  first  problem.  The  three  jealous 
husbands  have  been  traced  back  to  a  MS.  of  the  thirteenth 
century,  which  represents  two  German  youths,  Firri  and 
Tyrri,  proposing  problems  to  each  other.2  The  MS.  contains 
also  the  following :  Firri  says  :  "  There  were  three  brothers  in 
Cologne,  having  nine  vessels  of  wine.  The  first  vessel  con- 
tained one  quart  (amam),  the  second  2,  the  third  3,  the 
fourth  4,  the  fifth  5,  the  sixth  6,  the  seventh  7,  the  eighth 
8,  the  ninth  9.  Divide  the  wine  equally  among  the  three 
brothers,  without  mixing  the  contents  of  the  vessels."  This 

1  PEACOCK,  p.  473. 

2  DR.   S.  GUNTHER,    Geschichte   des  mathematischen   Unterrichts   im 
deutschen  Mittelalter,  Berlin,  1887,  p.  35. 


MODERN   TIMES  221 

question  is  closely  related  to  the  third  problem  given  above, 
since  it  gives  rise  to  the  following  magic  square  demanded  by 
that  problem. 

Magic  squares  were  known  to  the  Arabs  and,  perhaps,  to 
the  Hindus.  To  the  Byzantine  writer,  Moschopulus,  who  lived 
in  Constantinople  in  the  early  part  of  the  fifteenth 
century,  appears  to  be  due  the  introduction  into 
Europe  of  these  curious  and  ingenious  products 
of  mathematical  thought.  Mediaeval  astrologers 
believed  them  to  possess  mystical  properties  and 
when  engraved  on  silver  plate  to  be  a  charm  against  plague.1 
The  first  complete  magic  square  which  has  been  discovered 
in  the  Occident  is  that  of  the  German  painter,  Albrecht  Dtirer, 
found  on  his  celebrated  wood-engraving,  "  Melancholia." 

Of  interest  is  the  following  problem,  given  in  Kersey's 
Wingate :  "  15  Christians  and  15  Turks,  being  at  sea  in  one 
and  the  same  ship  in  a  terrible  storm,  and  the  pilot  declaring 
a  necessity  of  casting  the  one  half  of  those  persons  into  the 
sea,  that  the  rest  might  be  saved;  they  all  agreed,  that  the 
persons  to  be  cast  away  should  be  set  out  by  lot  after  this 
manner,  viz.  the  30  persons  should  be  placed  in  a  round  form 
like  a  ring,  and  then  beginning  to  count  at  one.  of  the  pas- 
sengers, and  proceeding  circularly,  every  ninth  person  should 
be  cast  into  the  sea,  until  of  the  30  persons  there  remained 
only  15.  The  question  is,  how  those  30  persons  ought  to  be 
placed,  that  the  lot  might  infallibly  fall  upon  the  15  Turks 
and  not  upon  any  of  the  15  Christians  ? "  Kersey  lets  the 
letters  a,  e,  i,  o,  u  stand,  respectively,  for  1,  2,  3,  4,  5,  and 

gives  the  verse 

From  numbers'  aid  and  art, 
Never  will  fame  depart. 

1  For  the  history  of  Magic  Squares,  see  GUNTHER,  Vermischte  Unter- 
suchungen,  Ch.  IV.  Their  theory  is  developed  in  the  article  "Magic 
Squares"  in  JOHNSON'S  Universal  Cyclopaedia. 


222  A    HISTORY   OF   MATHEMATICS 

The  vowels  in  these  lines,  taken  in  order,  indicate  alter- 
nately the  number  of  Christians  and  Turks  to  be  placed  to- 
gether ;  i.e.,  take  o  =  4  Christians,  then  u  =  5  Turks,  then 
e  =  2  Christians,  etc.  Bachet  de  Meziriac,  Tartaglia,  and 
Cardan  give  each  different  verses  to  represent  the  rule.  Ac- 
cording to  a  story  related  by  Hegesippus,1  the  famous  historian 
Josephus,  the  Jew,  while  in  a  cave  with  40  of  his  country- 
men, who  had  fled  from  the  conquering  Romans  at  the  siege 
of  Jotapata,  preserved  his  life  by  an  artifice  like  the  above. 
Rather  than  be  taken  prisoners,  his  countrymen  resolved  to 
kill  one  another.  Josephus  prevailed  upon  them  to  proceed 
by  lot  and  managed  it  so  that  he  and  one  companion  remained. 
Both  agreed  to  live. 

The  problem  of  the  15  Christians  and  15  Turks  has  been 
called  by  Cardan  Ludus  Joseph,  or  JosepH's  Play.  It  has 
been  found  in  a  French  work  of  1484  written  by  Nicolas 
Chuquet2  and  in  MSS.  of  the  twelfth,  eleventh,  and  tenth 
centuries.3  Daniel  Adams  gives  in  his  arithmetic  the  follow- 
ing stanza : 

"  As  I  was  going  to  St.  Ives, 

I  met  seven  wives. 

Every  wife  had  seven  sacks ; 

Every  sack  had  seven  cats  ; 

Every  cat  had  seven  kits  : 

Kits,  cats,  sacks,  and  wives, 

How  many  were  going  to  St.  Ives  ?  " 

Compare  this  with  Fibonaci's  "  Seven  old  women  go  to  Rome," 
etc.,  and  with  the  problem  in  the  Ahmes  papyrus,  and  we 
perceive  that  of  all  problems  in  "mathematical  recreations" 
this  is  the  oldest. 

Pleasant  and  diverting  questions  were  introduced  into  some 
English  arithmetics  of  the  latter  part  of  the  seventeenth  and 

1  De  Bello  Judaico,  etc.,  III.,  Ch.  15.       2  CANTOR,  Vol.  II.,  p.  362. 
3  M.  CURTZE,  in  Biblio.  Mathem.,  1894,  p.  116  and  1895,  pp.  34-36. 


MODERN   TIMES  223 

of  the  eighteenth  centuries.  In  Germany  this  subject  found 
entrance  into  arithmetic  during  the  sixteenth  century.  Its 
aim  was  to  make  arithmetic  more  attractive.  In  the  seven- 
teenth century  a  considerable  number  of  German  books  were 
wholly  devoted  to  this  subject.1 

Certain  writers  have  found  amusement  in  speculations  over 
the  origin  of  the  $.  Over  a  dozen  different  theories  have  been 
advanced,  but  no  serious  effort  was  made  to  verify  any  of 
them.  A  careful  study  of  manuscripts,  made  recently,2  has 
shown  that  the  $  is  the  lineal  descendant  of  the  Spanish 
abbreviation  p8  for  "  pesos,"  that  the  change  from  the  flo- 
rescent  p8  to  $  was  made  about  1775  by  English-Americans 
who  came  in  business  relations  with  Spanish-Americans.  The 
earliest  known  printed  $  occurs  in  an  arithmetic,  Chauncey 
Lee's  American  Accomptant,  published  in  1797  at  Lansing- 
burgh. 

1  WlLDERMUTH. 

2  See  F.  CAJORI  in  the  Popular  Science  Monthly,  December,  1912, 
pp.  521-630;  Science,  N.  S.,  Vol.  38,  1913,  pp.  848-850. 


ALGEBRA 


The  Renaissance 

ONE  of  the  great  steps  in  the  development  of  algebra  during 
the  sixteenth  century  was  the  algebraic  solution  of  cubic 
equations.  The  honour  of  this  remarkable  feat  belongs  to 
the  Italians.1  The  first  successful  attack  upon  cubic  equa- 
tions was  made  by  Scipio  Ferro  (died  in  1526),  professor 
of  mathematics  at  Bologna.  He  solved  cubics  of  the  form 
y?  +  mx  =  n,  but  nothing  more  is  known  of  his  solution  than 
that  he  taught  it  to  his  pupil  Floridus  in  1505.  It  was  the 
practice  in  those  days  and  during  centuries  afterwards  for 
teachers  to  keep  secret  their  discoveries  or  their  new  methods 
of  treatment,  in  order  that  pupils  might  not  acquire  this 
knowledge,  except  at  their  own  schools,  or  in  order  to  secure 
an  advantage  over  rival  mathematicians  by  proposing  prob- 
lems beyond  their  reach.  This  practice  gave  rise  to  many 
disputes  on  the  priority  of  inventions.  One  of  the  most 
famous  of  these  quarrels  arose  in  connection  with  the  dis- 
covery of  cubics,  between  Tartaglia  and  Cardan.  In  1530 
one  Colla  proposed  to  Tartaglia  several  problems,  one  leading 
to  the  equation  x*+px2  =  q.  The  latter  found  an  imperfect 
method  of  resolving  this,  made  known  his  success,  but  kept 

1  The  geometric  solution  had  been  given  previously  by  the  Arabs. 

224 


THE   RENAISSANCE  225 

Ms  solution  secret.  This  led  Ferro's  pupil  Eloridus  to  pro- 
claim his  knowledge  of  how  to  solve  x3  +  mx  =  n.  Tartaglia 
challenged  him  to  a  public  contest  to  take  place  Feb.  22,  1535. 
Meanwhile  he  worked  hard,  attempting  to  solve  other  cases 
of  cubic  equations,  and  finally  succeeded,  ten  days  before  the 
appointed  date,  in  mastering  the  case  $  =  mx  -+-  n.  At  the 
contest  each  man  proposed  30  problems.  The  one  who  should 
be  able  to  solve  the  greater  number  within  fifty  days  was  to 
be  the  victor.  Tartaglia  solved  his  rival's  problems  in  two 
hours ;  Floridus  could  not  solve  any  of  Tartaglia's.  Thence- 
forth Tartaglia  studied  cubic  equations  with  a  will,  and  in 
1541  he  was  in  possession  of  a  general  solution.  His  fame 
began  to  spread  throughout  Italy.  It  is  curious  to  see  what 
interest  the  enlightened  public  took  in  contests  of  this  sort. 
A  mathematician  was  honoured  and  admired  for  his  ability. 
Tartaglia  declined  to  make  known  his  method,  for  it  was  his 
aim  to  write  a  large  work  on  algebra,  of  which  the  solution 
of  cubics  should  be  the  crowning  feature.  But  a  scholar  of 
Milan,  named  Hieronimo  Cardano  (1501-1576),  after  many 
solicitations  and  the  most  solemn  promises  of  secrecy,  suc- 
ceeded in  obtaining  from  Tartaglia  the  method.  Cardan 
thereupon  inserted  it  in  a  mathematical  work,  the  Ars  Magna, 
then  in  preparation,  which  he  published  in  1545.  This  breach 
of  promise  almost  drove  Tartaglia  mad.  His  first  step  was 
to  write  a  history  of  his  invention,  but  to  completely  annihi- 
late Cardan,  he  challenged  him  and  his  pupil  Ferrari  to  a 
contest.  Tartaglia  excelled  in  his  power  of  solving  problems, 
but  was  treated  unfairly.  The  final  outcome  of  all  this  was 
that  the  man  to  whom  we  owe  the  chief  contribution  to 
algebra  made  in  the  sixteenth  century  was  forgotten,  and  the 
discovery  in  question  went  by  the  name  of  Cardan's  solution. 
Cardan  was  a  good  mathematician,  but  the  association  of 
his  name  with  the  discovery  of  the  solution  of  cubics  is  a 
Q 


226  A    HISTORY   OF   MATHEMATICS 

gross  historical  error  and  a  great  injustice  to  the  genius  of 
Tartaglia. 

The  success  in  resolving  cubics  incited  mathematicians  to 
extraordinary  efforts  toward  the  solution  of  equations  of 
higher  degrees.  The  solution  of  equations  of  the  fourth  degree 
was  effected  by  Cardan's  pupil,  Lodovico  Ferrari.  Cardan 
had  the  pleasure  of  publishing  this  brilliant  discovery  in  the 
Ars  Magna  of  1545.  Ferrari's  solution  is  sometimes  ascribed 
to  Bombelli,  who  is  no  more  the  discoverer  of  it  than  Cardan 
is  of  the  solution  called  by  his  name.  For  the  next  three 
centuries  algebraists  made  innumerable  attempts  to  discover 
algebraic  solutions  of  equations  of  higher  degree  than  the 
fourth.  It  is  probably  no  great  exaggeration  to  say  that  every 
ambitious  young  mathematician  sooner  or  later  tried  his 
skill  in  this  direction.  At  last  the  suspicion  arose  that  this 
problem,  like  the  ancient  ones  of  the  quadrature  of  the  circle, 
duplication  of  the  cube,  and  trisection  of  an  angle,  did  not 
admit  of  the  kind  of  solution  sought.  To  be  sure,  particu- 
lar forms  of  equations  of  higher  degrees  could  be  solved  satis- 
factorily. For  instance,  if  the  coefficients  are  all  numbers, 
some  method  like  that  of  Vieta,  Newton,  or  Homer,  always 
enables  the  computor  to  approximate  to  the  numerical  values 
of  the  roots.  But  suppose  the  coefficients  are  letters  which 
may  stand  for  any  number  whatever,  and  that  no  relation  is 
assumed  to  exist  between  these  coefficients,  then  the  problem 
assumes  more  formidable  aspects.  Finally  it  occurred  to  a 
few  mathematicians  that  it  might  be  worth  while  to  try  to 
prove  the  impossibility  of  solving  the  quintic  algebraically; 
that  is,  by  radicals.  Thus,  an  Italian  physician,  Paolo  Ruffini 
(1765-1822),  printed  proofs  of  their  insolvability,1  but  these 
proofs  were  declared  inconclusive  by  his  countryman  Malfatti. 

1  See  H.  BURKHARDT,  "Die  Anfange  der  Gruppentheorie  und  Paolo 
Ruffini"  in  Zeitschr.f.  Math.  u.  Physik,  Suppl.,  1892. 


THE   RENAISSANCE  227 

Later  a  brilliant  young  Norwegian,  Niels  Henrik  Abel  (1802- 
1829),  succeeded  in  establishing  by  rigorous  proof  that  the 
general  algebraic  equation  of  the  fifth  or  of  higher  degrees 
cannot  be  solved  by  radicals.1  A  modification  of  AbePs  proof 
was  given  by  Wantzel.2 

Keturning  to  the  Renaissance  it  is  interesting  to  observe, 
that  Cardan  in  his  works  takes  notice  of  negative  roots  of  an 
equation  (calling  them  fictitious)  while  the  positive  roots  are 
called  real),  and  discovers  all  three  roots  of  certain  numerical 
cubics  (no  more  than  two  roots  having  ever  before  been 
found  in  any  equation).  While  in  his  earlier  writings  he 
rejects  imaginary  roots  as  impossible,  in  the  Ars  Magna  he 
exhibits  great  boldness  of  thought  in  solving  the  problem,  to 
divide  10  into  two  parts  whose  product  is  40,  by  finding  the 
answers  5+V— 15  and  5— V— 15,  and  then  multiplying 
them  together,  obtaining  25  -+•  15  =  40.3  Here  for  the  first 
time  we  see  a  decided  advance  on  the  position  taken  by  the 
Hindus.  Advanced  views  on  imaginaries  were  held  also  by 
Raphael  Bombelli,  of  Bologna,  who  published  in  1572  an  alge- 
bra in  which  he  recognized  that  the  so-called  irreducible  case 
in  cubics  gives  real  values  for  the  roots. 

It  may  be  instructive  to  give  examples  of  the  algebraic 
notation  adopted  in  Italy  in  those  days.4 

1  See  CRELLE'S  Journal,  I.,  1826. 

2  Wantzel's  proof,  translated  from  Serret's  Cours  d'Algebre  Superieure, 
was  published  in  Vol.  IV.,  p.  65,  of  the  Analyst,  edited  by  JOEL  E.  HEN- 
DRICKS  of  Des  Moines.    While  the  quintic  cannot  be  solved  by  radicals,  a 
transcendental  solution,  involving  elliptic  integrals,  was  given  by  Hermite 
(in  the  Compt.  Eend.,  1858,  1865,  1866)  and  by  Kronecker  in  1858.    A 
translation  of  the  solution  by  elliptic  integrals,  taken  from  Briot  and 
Bouquet's  Theory  of   Elliptic  Functions,  is  likewise  published  in   the 
Analyst,  Vol.  V.,  p.  161.  8  CANTOR,  II. ,  508. 

4  CANTOR,  II.,  320  ;  MATTHIESSEN,  p.  368  ;  the  value  of  x  given  on  the 
following  page  is  the  solution  of  the  cubic  in  the  previous  line.  The 
u  V"  or  "  v  "  is  a  sign  of  aggregation  or  joint  root. 


228  A   HISTORY   OF   MATHEMATICS 

Pacioli  :     #  V  40  m  #  320,  V40  -  V320~. 

Cardan  :    Cubus  _p  6  rebus  sequalis  20,    x?  +  6  x  =  20, 
$.  v.  cu.  #.  108  p.  10  |  m  #.  v.  cu.  $.  108  m  10, 

l6  -  ^  V108  -  10~ 


The  Italians  were  in  the  habit  of  calling  the  unknown 
quantity  cosa,  "thing."  In  Germany  this  word  was  adopted 
as  early  as  the  time  of  John  Widmann  as  a  name  for  algebra  : 
he  speaks  of  the  "Regel  Algobre  oder  Cosse";  in  England 
this  new  name  for  algebra,  the  cossic  art,  gave  to  the  first 
English  work  thereon,  by  Robert  Recorde,  its  punning  title 
the  Whetstone  of  Witte,  truly  a  cos  ingenii.  The  Germans  made 
important  contributions  to  algebraic  notation.  The  +  and  — 
signs,  mentioned  by  us  in  the  history  of  arithmetic,  were,  of 
course,  introduced  into  algebra,  but  they  did  not  pass  into  gen- 
eral use  before  the  time  of  Vieta.  "  It  is  very  singular/7  says 
Hallam,  "that  discoveries  of  the  greatest  convenience,  and, 
apparently,  not  above  the  ingenuity  of  a  village  schoolmaster, 
should  have  been  overlooked  by  men  of  extraordinary  acuteness 
like  Tartaglia,  Cardan,  and  Ferrari  ;  and  hardly  less  so  that,  by 
dint  of  that  acuteness,  they  dispensed  with  the  aid  of  these 
contrivances  in  which  we  suppose  that  so  much  of  the  utility 
of  algebraic  expression  consists."  Another  important  symbol 
introduced  by  the  Germans  is  the  radical  sign.  In  a  manu- 
script published  some  time  in  the  fifteenth  century,  a  dot 
placed  before  a  number  is  made  to  signify  the  extraction  of  a 
root  of  that  number.  Cliristoff  Rudolff,  who  wrote  the  earliest 
text-book  on  algebra  in  the  German  language  (printed  1525), 
remarks  that  "the  radix  quadrata  is,  for  brevity,  designated 
in  his  algorithm  with  the  character  ^/>  as  -y/4."  Here  the  dot 
found  in  the  manuscript  has  grown  into  a  symbol  much  like 
our  own.  With  him  VVV  and  VV  stand  for  the  cube  and 


THE   RENAISSANCE  229 

the  fourth  roots.  The  symbol  y'  was  used  by  Michael  Stifel 
(1486  ?-1567),  who,  in  1553,  brought  out  a  second  edition  of 
RudolfF  s  Cossj  containing  rules  for  solving  cubic  equations, 
derived  from  the  writings  of  Cardan.  Stifel  ranks  as  the 
greatest  German  algebraist  of  the  sixteenth  century.  He  was 
educated  in  a  monastery  at  Esslingen,  his  native  place,  and 
afterwards  became  a  Protestant  minister.  Study  of  the 
significance  of  mystic  numbers  in  Revelation  and  in  Daniel 
drew  him  to  mathematics.  He  studied  German  and  Italian 
works,  and  in  1544  published  a  Latin  treatise,  the  Arithmetica 
Integra,  given  to  arithmetic  and  algebra.  Therein  he  observes 
the  advantage  in  letting  a  geometric  series  correspond  to  an 
arithmetic  series,  remarking  that  it  is  possible  to  elaborate  a 
whole  book  on  the  wonderful  properties  of  numbers  depending 
on  this  relation.  He  here  makes  a  close  approach  to  the  idea 
of  a  logarithm.  He  gives  the  binomial  coefficients  arising  in 
the  expansion  of  (a  +  b)  to  powers  below  the  18th,  and  uses 
these  coefficients  in  the  extraction  of  roots.  German  notations 
are  illustrated  by  the  following  : 

Regiomontanus  :     16  census  et  2000  sequales  680  rebus, 

16  a2  +  2000  -680  x. 

Stifel:  V$-V22°-4-  V68>  V-V20-  4-^/8. 


The  greatest  French  algebraist  of  the  sixteenth  century  was 
Franciscus  Vieta  (1540-1603).  He  was  a  native  of  Poitou  and 
died  in  Paris.  He  was  educated  a  lawyer  ;  his  manhood  he 
spent  in  public  service  under  Henry  III.  and  Henry  IV.  To 
him  mathematics  was  a  relaxation.  Like  Napier,  he  does  not 
profess  to  be  a  mathematician.  During  the  war  against  Spain, 
he  rendered  service  to  Henry  IV.  by  deciphering  intercepted 
letters  written  in  a  cipher  of  more  than  500  characters  with 
variable  signification,  and  addressed  by  the  Spanish  court  to 


230  A   HISTORY   OF   MATHEMATICS 

their  governor  of  Netherlands.  The  Spaniards  attributed  the 
discovery  of  the  key  to  magic.  Vieta  is  said  to  have  printed 
all  his  works  at  his  own  expense,  and  to  have  distributed  them 
among  his  friends  as  presents.  His  In  Artem  Analyticam 
Isagoge,  Tours,  1591,  is  the  earliest  work  giving  a  symbolical 
treatment  of  algebra.  Not  only  did  he  improve  the  algebra 
and  trigonometry  of  his  time,  but  he  applied  algebra  to  geom- 
etry to  a  larger  extent,  and  in  a  more  systematic  manner,  than 
had  been  done  before  him.  He  gave  also  the  trigonometric 
solution  of  Cardan's  irreducible  case  in  cubics. 

In  'the  solution  of  equations  Vieta  persistently  employed 
the  principle  of  reduction  and  thereby  introduced  a  uniformity 
of  treatment  uncommon  in  his  day.  He  reduces  affected  quad- 
ratics to  pure  quadratics  by  making  a  suitable  substitution 
for  the  removal  of  the  term  containing  x.  Similarly  for  cubics 
and  biquadratics.  Vieta  arrived  at  a  partial  knowledge  of 
the  relations  existing  between  the  coefficients  and  the  roots 
of  an  equation.  Unfortunately  he  rejected  all  except  positive 
roots,  and  could  not,  therefore,  fully  perceive  the  relations 
in  question.  His  nearest  approach  to  complete  recognition  of 
the  facts  is  contained  in  the  statement  that  the  equation 
x3  —  (u  +  v  -f-  w)x2  +  (uv  -f  vw  +  wu)x  —  uvw  =  0  has  the  three 
roots  u,  v,  w.  For  cubics,  this  statement  is  perfect,  if  u,  v,  w,  are 
allowed  to  represent  any  numbers.  But  Vieta  is  in  the  habit 
of  assigning  to  letters  only  positive  values,  so  that  the  passage 
really  means  less  than  at  first  sight  it  appears  to  do.1  As 
early  as  1558.  Jacques  Peletier  (1517-1582),  a  French  text-book 
writer  on  algebra  and  geometry,  observed  that  the  root  of  an 
equation  is  a  divisor  of  the  last  term.  Broader  views  were 
held  by  Albert  Girard  (1590-1633),  a  noted  Flemish  mathema- 
tician, who  in  1629  issued  his  Invention  nouvelle  en  Valgebre, 

i  HANREL,  p.  379. 


THE   RENAISSANCE  231 

He  was  the  first  to  understand  the  use  of  negative  roots  in  the 
solution  of  geometric  problems.  He  spoke  of  imaginary  quan- 
tities, and  inferred  by  induction  that  every  equation  has  as 
many  roots  as  there  are  units  in  the  numbe*r  expressing  its 
degree.  He  first  showed  how  to  express  the  sums  of  their 
products  in  terms  of  the  coefficients.  The  sum  of  the  roots, 
giving  the  coefficient  of  the  second  term  with  the  sign  changed, 
he  called  the  premiere  faction;  the  sum  of  the  products  of 
the  roots,  two  and  two,  giving  the  coefficient  of  the  third  term, 
he  called  deuxfeme  faction,  etc.  In  case  of  the  equation 
x4— 4#+3  =  0,  he  gives  the  roots  a?i=l,  a?2  =  l,  xs  =  —  1  +  V— 2, 
x4  =  —  1  —  V— 2,  and  then  states  that  the  imaginary  roots  are 
serviceable  in  showing  the  generality  of  the  law  of  formation 
of  the  coefficients  from  the  roots.1  Similar  researches  on  the 
theory  of  equations  were  made  in  England  independently  by 
Thomas  Harriot  (1560-1621).  His  posthumous  work,  the 
Artis  Analyticce  Praxis,  1631,  was  written  long  before  Girard's 
Invention,  though  published  after  it.  Harriot  discovered  the 
relations  between  the  roots  and  the  coefficients  of  an  equation 
in  its  simplest  form.  This  discovery  was  therefore  made 
about  the  same  time  by  Harriot  in  England,  and  by  Vieta 
and  Girard  on  the  continent.  Harriot  was  the  first  to  decom- 
pose equations  into  their  simple  factors,  but  as  he  failed  to 
recognize  imaginary,  or  even  negative  roots,  he  failed  to  prove 
that  every  equation  could  be  thus  decomposed. 

Harriot  was  the  earliest  algebraist  of  England.  After  grad- 
uating from  Oxford,  he  resided  with  Sir  Walter  Raleigh  as 
his  mathematical  tutor.2  Raleigh  sent  him  to  Virginia  as 
surveyor  in  1585  with  Sir  Richard  Grenville's  expedition. 
After  his  return,  the  following  year,  he  published  "A  Brief 
and  True  Report  of  the  New-found  Land  of  Virginia,"  which 

1  CANTOR,  II.,  788.  2  Dictionary  of  National  Biography. 


232  A   HISTORY   OF   MATHEMATICS 

excited  much  notice  and  was  translated  into  Latin.  Among 
the  mathematical  instruments  by  which  the  wonder  of  the 
Indians  was  aroused,  Harriot  mentions  "a  perspective  glass 
whereby  was  showed  many  strange  sights." 1  About  this 
time,  Henry,  Earl  of  Northumberland,  became  interested  in 
Harriot.  Admiring  his  affability  and  learning,  he  allowed 
Harriot  a  life-pension  of  £  300  a  year.  In  1606  the  Earl  was 
committed  to  the  Tower,  but  his  three  mathematical  friends, 
Harriot,  Walter  Warner,  and  Thomas  Hughes,  "the  three 
magi  of  the  Earl  of  Northumberland,"  frequently  met  there, 
and  the  Earl  kept  a  handsome  table  for  them.  Harriot  was 
in  poor  health,  which  explains,  perhaps,  his  failure  to  complete 
and  publish  his  discoveries. 

We  next  summarize  the  views  regarding  the  negative  and 
imaginary,  held  by  writers  of  the  sixteenth  century  and 
the  early  part  of  the  seventeenth.  Cardan's  "pure  minus" 
and  his  views  on  imaginaries  were  in  advance  of  his  age. 
Until  the  beginning  of  the  seventeenth  century  mathemati- 
cians dealt  exclusively  with  positive  quantities.  Pacioli  says 
that  "minus  times  minus  gives  plus,"  but  applies  this  only 
to  the  formation  of  the  product  of  (a  —  6)(c  —  d).  Purely 
negative  quantities  do  not  appear  in  his  work.  The  German 
"  Cossist,"  Rudolff,  knows  only  positive  numbers  and  positive 
roots,  notwithstanding  his  use  of  the  signs  +  and  — .  His 
successor,  Stifel,  speaks  of  negative  numbers  as  "less  than 
nothing,"  also  as  "absurd  numbers,"  which  arise  when  real 
numbers  above  zero  are  subtracted  from  zero.2  Harriot  is 
the  first  who  occasionally  places  a  negative  term  by  itself  on 
one  side  of  an  equation.  Yieta  knows  only  positive  numbers, 

1  Harriot  was  an  astronomer  as  well  as  mathematician,  and  he  "  applied 
the  telescope  to  celestial  purposes  almost  simultaneously  with  Galileo." 
His  telescope  magnified  up  to  50  times.     See  the  Die.  of  Nat.  Biography. 

2  CANTOR,  II.,  4te. 


THE   RENAISSANCE  233 

but  Girard  had  advanced  views,  both  on  negatives  and  imagi- 
naries.  Before  the  seventeenth  century,  the  majority  of  the 
great  European  algebraists  had  not  quite  risen  to  the  views 
taught  by  the  Hindus.  Only  a  few  can  be  said,  like  the 
Hindus,  to  have  seen  negative  roots ;  perhaps  all  Europeans, 
like  the  Hindus,  did  not  approve  of  the  negative.  The  full 
interpretation  and  construction  of  negative  quantities  and  the 
systematic  use  of  them  begins  with  Rent,  Descartes  (1596- 
1650),  but  after  him  erroneous  views  respecting  them  appear 
again  and  again.  In  fact,  not  until  the  middle  of  the  nine- 
teenth century  was  the  subject  of  negative  numbers  properly 
explained  in  school  algebras.  The  question  naturally  arises, 
why  was  the  generalization  of  the  concept  of  number,  so  as 
to  include  the  negative,  such  a  difficult  step  ?  The  answer 
would  seem  to  be  this  :  Negative  numbers  appeared  "  absurd  " 
or  "fictitious,"  so  long  as  mathematicians  had  not  hit  upon 
a  visual  or  graphical  representation  of  them.  The  Hindus  early 
saw  in  "opposition  of  direction"  on  a  line  an  interpretation 
of  positive  and  negative  numbers.  The  ideas  of  "  assets  "  and 
"  debts "  offered  to  them  another  explanation  of  their  nature. 
In  Europe  full  possession  of  these  ideas  was  not  acquired 
before  the  time  of  Girard  and  Descartes.  To  Stifel  is  due  the 
absurd  expression,  negative  numbers  are  "less  than  nothing." 
It  took  about  300  years  to  eliminate  this  senseless  phrase  from 
mathematical  language. 

History  emphasizes  the  importance  of  giving  graphical 
representations  of  negative  numbers  in  teaching  algebra. 
Omit  all  illustration  by  lines,  or  by  the  thermometer,  and 
negative  numbers  will  be  as  absurd  to  modern  students  as 
they  were  to  the  early  algebraists. 

In  the  development  of  symbolic  algebra  great  services  were 
rendered  by  Vieta.  Epoch-making  was  the  practice  intro- 
duced by  him  of  denoting  general  or  indefinite  expressions 


234  A    HISTORY   OF   MATHEMATICS 

by  letters  of  the  alphabet.  To  be  sure,  Stifel,  Cardan,  and 
others  used  letters  before  him,  but  Vieta  first  made  them  an 
essential  part  of  algebra.  The  new  symbolic  algebra  was 
called  by  him  logistica  speciosa  in  opposition  to  the  old  logistica 
numerosa.  By  his  notation  a3  -f-  3  a*b  +  3  ab2  -f-  bs  =  (a  +  b)3 
was  written  "  a  cubus  +  b  in  a  quadr.  3  -f-  a  in  b  quadr.  3  -f-  & 
cubo  aequalia  a  +  b  cubo."  The  vinculum  was  introduced  by 
him  as  a  sign  of  aggregation.  Parentheses  first  occur  with 
Girard.  In  numerical  equations  the  unknown  quantity  was  de- 
noted by  N9  its  square  by  Q,  and  its  cube  by  C.  Illustrations : * 

Vieta:         1(7-8  Q  +  16  JVsequ.  40,  tf-8x>  +  16a;  =  40. 
Vieta :         A  cubus  +  B  piano  3  in  A, 

sequari  Z  solido  2,  a?  -+-  3  bx  =  2  c. 

Girard:      1  ® »  13  ©  + 12,  a-3  =  13  a  +  12. 

Descartes :  a?*  +px  +  q  ao  0,  ic3  -f  px  +  g  =  0. 

Our  sign  of  equality,  =,  is  due  to  Eecorde.  Harriot  adopted 
small  letters  of  the  alphabet  in  place  of  the  capitals  used  by 
Vieta.  Harriot  writes  a3  —  3  ab2  —  2  c3  thus :  aaa  —  3  bba 
=  2  ccc.  The  symbols  of  inequality,  >  and  <,  were  intro- 
duced by  him.  William  Oughtred  (1574-1660)  introduced 
X  as  a  symbol  of  multiplication,  :  :  for  proportion,  ~  for  dif- 
ference. In  his  Clavis,  1631,  he  writes  A10  thus,  Aqqcc ;  and 
120  A7E*  thus,  120  AqqcEc.  The  sign  :  for  ratio  was  used  by 
Vincent  Wing  in  1651 ;  the  symbol  oc  for  variation,  by  William 
Emerson  in  1768. 

The  Last  Three  Centuries 

The  first  steps  toward  the  building  up  of  our  modern 
theory  of  exponents  and  our  exponential  notation  were  taken 
by  Simon  Stevin  (1548-1620)  of  Bruges  in  Belgium.  Oresme's 
previous  efforts  in  this  direction  remained  wholly  unnoticed, 

1  MATTHIESSEN,  pp.  270,  371. 


THE   LAST   THREE   CENTURIES  235 

but  Stevin's  innovations,  though  neglected  at  first,  are  a 
permanent  possession.  His  exponential  notation  grew  in 
connection  with  his  notation  for  decimal  fractions.  Denoting 
the  unknown  quantity  by  Q>  ne  places'  within  the  circle  the 
exponent  of  the  power.  Thus  ®,  @,  (3)  signify  #,  a2,  a?.  He 
extends  his  notation  to  fractional  exponents.  (J),  (T),  (J),  mean 
o$,  x*,  x$.  He  writes  Sxyz2  thus  3©  M  sec@  M  ter®,  where 
M  means  multiplication;  sec,  second;  ter,  third  unknown 
quantity.  The  Q  for  x  was  adopted  by  Girard.  Stevin's 
great  independence  of  mind  is  exhibited  in  his  condemnation 
of  such  terms  as  "  sursolid,"  or  numbers  that  are  "  absurd," 
"irrational,"  "irregular,"  "surd."  He  shows  that  all  num- 
bers are  equally  proper  expressions  of  some  length,  or  some 
power  of  the  same  root.  He  also  rejects  all  compound  ex- 
pressions, such  as  "square-squared,"  "cube-squared,"  and 
suggests  that  they  be  named  by  their  exponents  the  "  fourth," 
"fifth"  powers.  Stevin's  symbol  for  the  unknown  failed  to 
be  adopted,  but  the  principle  of  his  exponential  notation  has 
survived.  The  modern  formalism  took  its  shape  with  Des- 
cartes. In  his  Geometry,  1637,  he  uses  the  last  letters  of  the 
alphabet,  x  in  the  first  place,  then  the  letters  y,  z  to  designate 
unknown  quantities ;  while  the  first  letters  of  the  alphabet  are 
made  to  stand  for  known  quantities.  Our  exponential  nota- 
tion, a4,  is  found  in  Descartes ;  however,  he  does  not  use  gen- 
eral exponents,  like  an,  nor  negative  and  fractional  ones.  In 
this  last  respect  he  did  not  rise  to  the  ideas  of  Stevin.  In  case 
of  radicals  he  does  not  indicate  the  root  by  indices,  but  in 
case  of  cube  root,  for  instance,  uses  the  letter  (7,  thus, 

VC+T? = A/p.1 

Of  the  early  notations  for  evolution,  two  have  come  down 
to  our  time,  the  German  radical  sign  and  Stevin's  fractional 

1  CANTOR,  II,  794. 


236  A  HISTORY   OF   MATHEMATICS 

exponents.  Modern  pupils  have  to  learn  the  algorithm  for 
both  notations ;  they  must  learn  the  meaning  of  -\/a2,  and  also 
of  its  equal  a*.  It  is  a  great  pity  that  this  should  be  so.  The 
operations  with  fractional  exponents  are  not  always  found  easy, 
and  the  rules  for  radicals  are  always  pronounced  "  hard."  By 
learning  both,  the  progress  of  the  pupil  is  unnecessarily  re- 
tarded. Of  the  two,  the  exponential  notation  is  immeasurably 
superior.  Radicals  appear  only  in  evolution.1  Exponents,  on 
the  other  hand,  apply  to  both  involution  and  evolution ;  with 
them  all  operations  and  simplifications  are  effected  with  com- 
parative ease.  In  case  of  radicals,  what  a  gain  it  would  be,  if 
we  could  burst  the  chains  which  tie  us  to  the  past ! 

Descartes  enriched  the  theory  of  equations  with  a  theorem 
which  goes  by  the  name  of  "  rule  of  signs."  By  it  are  deter- 
mined the  number  of  positive  and  negative  roots  of  an  equation : 
the  equation  may  have  as  many  -f-  roots  as  there  are  variations 
in  sign,  and  as  many  —  roots  as  there  are  permanences  in  sign. 
Descartes  was  accused  by  Wallis  of  availing  himself,  without 
acknowledgment,  of  Harriot's  theory  of  equations,  particularly 
his  mode  of  generating  equations ;  but  there  seems  to  be  no 
good  ground  for  the  charge.  Wallis  claimed,  moreover,  that 
Descartes  failed  to  notice  that  the  rule  breaks  down  in  case  of 
imaginary  roots,  but  Descartes  does  not  say  that  the  equation 
always  has,  but  that  it  may  have,  so  many  roots.  It  is  true 
that  Descartes  does  not  consider  the  case  of  imaginaries 
directly ;  but  further  on,  in  his  Geometry,  he  gives  ample  evi- 
dence of  his  ability  to  handle  the  case  of  imaginaries. 

1  In  connection  with  the  imaginary,  V—  1,  the  radical  notation  is 
objectionable,  because  it  leads  students  and  even  authors  to  remark  that 
the  rules  of  operation  for  real  quantities  do  not  always  hold  for  imagi- 
naries, since  V  —  1  •  V—  1  does  not  equal  V+  1.  That  the  difficulty  is 
merely  one  of  notation  is  evident  from  the  fact  that  it  disappears  when 
we  designate  the  imaginary  unit  by  i.  Then  i  •  i  =  t2,  which,  by 
definition,  equals  —  1. 


THE   LAST   THREE   CENTURIES  237 

John  Wallis  (1616-1703)  was  an  English  mathematician  of 
great  originality.  He  was  educated  for  the  Church,  at  Cam- 
bridge, and  took  Holy  Orders,  but  in  1649  was  appointed 
Savilian  professor  of  geometry  at  Oxford.  He  advanced 
beyond  Kepler  by  making  more  extended  use  of  the  "  law  of 
continuity,"  applying  it  to  algebra,  while  Desargues  applied 
it  to  geometry.  By  this  law  Wallis  was  led  to  regard  the 
denominators  of  fractions  as  powers  with  negative  exponents. 
For  the  descending  geometrical  progression  ar2,  a?1,  a°,  if  con- 
tinued, gives  or1,  or2,  etc.;  which  is  the  same  thing  as  -,  — , 

etc.  The  exponents  of  his  geometric  series  are  in  the  arith- 
metical progression  2,  1,  0,  —  1,  —  2.  He  also  used  fractional 
exponents,  which  had  been  invented  long  before,  but  had 
failed  to  be  generally  introduced.  The  symbol  oo  for  infinity 
is  due  to  him.  In  1685  Wallis  published  an  Algebra  which 
has  long  been  a  standard  work  of  reference.  It  treats  of  the 
history,  theory,  and  practice  of  arithmetic  and  algebra.  The 
historical  part  is  unreliable  and  worthless,  but  in  other  respects 
the  work  is  a  masterpiece,  and  wonderfully  rich  in  material. 

The  study  of  some  results  obtained  by  Wallis  on  the  quad- 
rature of  curves  led  Newton  to  the  discovery  of  the  Binomial 
Theorem,  made  about  1665,  and  explained  in  a  letter  written 
by  Newton  to  Oldenburg  on  June  13, 1676.1  Newton's  reason- 
ing gives  the  development  of  (a  -f-  &)n,  whether  n  be  positive 
or  negative,  integral  or  fractional.  Except  when  n  is  a  positive 
integer,  the  resulting  series  is  infinite.  He  gave  no  regular 
proof  of  his  theorem,  but  verified  it  by  actual  multiplication. 
The  case  of  positive  integral  exponents  was  proved  by  James 
Bernoulli 2  (1654-1705),  by  the  doctrine  of  combinations.  A 

1  How  the  Binomial  Theorem  was  deduced  as  a  corollary  of  Wallis' a 
results  is  explained  in  Cajori's  History  of  Mathematics,  pp.  195,  196. 

2  Ars  Conjectm\di,  1713,  p.  89. 


238  A  HISTORY  OF  MATHEMATICS 

proof  for  negative  and  fractional  exponents  was  given  by 
Leonhard  Euler  (1707-1783).  It  is  faulty,  for  the  reason  that 
he  fails  to  consider  the  convergence  of  the  series  ;  nevertheless 
it  has  been  reproduced  in  elementary  text-books  even  of  recent 
years.1  A  rigorous  general  proof  of  the  Binomial  Theorem, 
embracing  the  case  of  incommensurable  and  imaginary  powers, 
was  given  by  Niels  Henrik  Abel.2  It  thus  appears  that  for  over 
a  century  and  a  half  this  fundamental  theorem  went  without 
adequate  proof.3 

Sir  Isaac  Newton  (1642-1727)  is  probably  the  greatest  mathe- 
matical mind  of  all  times.  Some  idea  of  his  strong  intuitive 
powers  may  be  obtained  from  the  fact  that  as  a  youth  lie 
regarded  the  theorems  of  ancient  geometry  as  self-evident 
truths,  and  that,  without  any  preliminary  study,  he  made 
himself  master  of  Descartes'  Geometry.  He  afterwards  re- 
garded this  neglect  of  elementary  geometry  as  a  mistake,  and 
once  expressed  his  regret  that  "  he  had  applied  himself  to  the 
works  of  Descartes  and  other  algebraic  writers  before  he  had 
considered  the  Elements  of  Euclid  with  the  attention  that  so 
excellent  a  writer  deserves."  During  the  first  nine  years  of 

1  For  the  history  of  Infinite  Series  see  REIFF,  Geschichte  der  Unend- 
lichen  Reihen,  Tubingen,  1889;   CANTOR,  III.,  53-94;  CAJORI,  pp.  334- 
339 ;  Teach,  and  Hist,  of  Math,  in  the  U.S.,  pp.  361-376. 

2  See  CRELLE,  I.,  1827,  or  (Entires  completes,  de  N.  H.  ABEL,  Chris- 
tiania,  1839,  I.,  66  et  seq. 

3  It  should  be  mentioned  that  the  beginnings  of  the  Binomial  Theorem 
for  positive  integral  exponents  are  found  very  early.     The  Hindus  and 
Arabs  used  the  expansions  of  (a  +  6)2  and  (a  4-  6)3  in  the  extraction  of 
square  and  cube  root.    Vieta  knew  the  expansion  of  (a  +  6)4.    But  these 
were  obtained  by  actual  multiplication,  not  by  any  law  of  expansion. 
Stifel  gave  the  coefficients  for  the  first  18  powers ;  Pascal  did  similarly 
in  his  "arithmetical  triangle"   (see  CANTOR,  II.,  685,  686).     Pacioli, 
Stevin,  Briggs,  and  others  also  possessed  something,  from  which  one 
would  think  the  Binomial  Theorem  could  have  been  derived  with  a  little 
attention,  "if  we  did  not  know  that  such  simple  relations  were  difficult 
to  discover"  (De  Morgan). 


THE   LAST   THREE   CENTURIES  239 

Ms  professorship  at  Cambridge  he  delivered  lectures  on  alge- 
bra. Over  thirty  years  after,  they  were  published,  in  1707,  by 
Mr.  Whiston  under  the  title,  Arithmetica  Uhiversalis.  They 
contain  new  and  important  results  on  the  theory  of  equations. 
His  theorem  on  the  sums  of  powers  of  roots  is  well  known. 
We  give  a  specimen  of  his  notation : 

a3  4-  2  aac  —  aab  —  3  abc  +  bbc. 

Elsewhere  in  his  works  he  introduced  the  system  of  literal 
indices.  The  Arithmetica  Uhiversalis  contains  also  a  large 
number  of  problems.  Here  is  one  (No.  50) :  "  A  stone  falling 
down  into  a  well,  from  the  sound  of  the  stone  striking  the 
bottom,  to  determine  the  depth  of  the  well."  He  leaves  his 
problems  with  the  remark  which  shows  that  methods  of  teach- 
ing secured  some  degree  of  attention  at  his  hands :  "  Hitherto 
I  have  been  solving  several  problems.  For  in  learning  the 
sciences  examples  are  of  more  use  than  precepts."  * 

1  Newton's  body  was  interred  in  Westminster  Abbey,  where  in  1731 
a  fine  monument  was  erected  to  his  memory.  In  cyclopaedias,  the 
statement  is  frequently  made  that  the  Binomial  Formula  was  engraved 
upon  Newton's  tomb,  but  this  is  very  probably  not  correct,  for  the 
following  reasons:  (1)  We  have  the  testimony  of  Dr.  Bradley,  the 
Dean  of  Westminster,  and  of  mathematical  acquaintances  who  have 
visited  the  Abbey  and  mounted  the  monument,  that  the  theorem  can 
not  be  seen  on  the  tomb  at  the  present  time.  Yet  all  Latin  inscriptions 
are  still  distinctly  readable.  (2)  None  of  the  biographers  of  Newton 
and  none  of  the  old  guide-books  to  Westminster  Abbey  mention  the 
Binomial  Formula  in  their  (often  very  full)  descriptions  of  Newton's 
tomb.  However,  some  of  them  say,  that  on  a  small  scroll  held  by  two 
winged  youths  in  front  of  the  half -recumbent  figure  of  Newton,  there  are 
some  mathematical  figures.  See  Neale's  Guide.  Brewster,  in  his  Life  of 
Sir  Isaac  Newton,  1831,  says  that  a  "converging  series"  is  there,  but 
this  does  not  show  now.  Brewster  would  surely  have  said  "Binomial 
Theorem  "  instead  of  "  converging  series  "  had  the  theorem  been  there. 
The  Binomial  Formula,  moreover,  is  not  always  convergent.  (3)  It 
is  important  to  remember  that  whatever  was  engraved  on  the  scroll 
could  not  be  seen  and  read  by  any  one,  unless  he  stood  on  a  chair  or 


240  A   HISTORY   OF   MATHEMATICS 

The  principal  investigators  on  the  solution  of  numerical 
equations  are  Vieta,  Newton,  Lagrange,  Joseph  Fourier,  and 
Horner.  Before  Vieta,  Cardan  applied  the  Hindu  rule 
of  "false  position"  to  cubics,  but  his  method  was  crude. 
Vieta  gave  a  process  which  is  laborious,  but  is  the  forerunner 
of  the  easy  methods  of  Newton  and  Horner.1  The  later 
changes  are  in  the  arrangement  of  the  work,  so  as  to  afford 
facility  and  security  in  the  evolution  of  the  root.  Horner's 
process  is  the  one  usually  taught.  William  George  Horner 
(1786-1837),  of  Bath,  the  son  of  a  Wesleyan  minister,  was 
educated  at  Kingswood  School,  near  Bristol.  His  method 
was  read  before  the  Royal  Society,  July  1,  1819.2  Almost 
precisely  the  same  method  was  given  by  the  Italian  Paolo 
Euffini  in  1804.  Neither  Ruffini  nor  Horner  knew  that  their 
method  had  been  given  by  the  Chinese  as  early  as  the  thir- 
teenth century.  De  Morgan  admired  Horner's  method  and  per- 
fected it  in  details  still  further.  It  was  his  conviction  that  it 
should  be  included  in  the  arithmetics ;  he  taught  it  to  his 
classes,  and  derided  the  examiners  at  Cambridge  who  ignored 
the  method.3  De  Morgan  encouraged  students  to  carry  out 

used  a  step-ladder.  Whatever  was  written  on  the  scroll  was,  therefore, 
not  noticed  by  transient  visitors.  The  persons  most  likely  to  examine 
everything  carefully  would  be  the  writers  of  the  guide-books  and  the 
biographers  —  the  very  ones  who  are  silent  on  the  Binomial  Theorem.  On 
the  other  hand,  such  a  writer  as  E.  Stone,  the  compiler  of  the  New  Math- 
ematical Dictionary,  London,  1743,  is  more  likely  to  base  his  statement 
that  the  theorem  is  "put  upon  the  monument,"  upon  hearsay.  (4)  We 
have  the  positive  testimony  of  a  most  accurate  writer,  Augustus  De 
Morgan,  that  the  theorem  is  not  inscribed.  Unfortunately  we  have  no- 
where seen  his  reasons  for  the  statement.  See  his  article  "  Newton  "  in 
the  Penny  or  the  English  Cyclopaedia.  See  also  our  article  in  the  Bull, 
of  the  Am.  Math.  Soc.,  I.,  1894,  pp.  52-54. 

1  Colo.  College  Publication,  General  Series  51  and  52,  1910. 

2  Dictionary  of  National  Biography. 

3  See  DE  MORGAN,  Budget  of  Paradoxes,  2d  Ed.,  1915,  Vol.  II.,  pp.  188, 
189. 


THE   LAST   THREE   CENTURIES  241 

long  arithmetical  computations  for  the  sake  of  acquiring  skill 
and  rapidity.  Thus,  one  of  his  pupils  solved  a?3  —  2  x  =  5  to 
103  decimal  places,  "  another  tried  150  places,  but  broke  down 
at  the  76th,  which  was  wrong." l  While,  in  our  opinion,  De 
Morgan  greatly  overestimated  the  value  of  Horner's  method 
to  the  ordinary  boy,  and,  perhaps, '  overdid  in  matters  of 
calculation,  it  is  certainly  true  that  in  America  teachers 
have  gone  to  the  other  extreme,  neglecting  the  art  of  rapid 
computation,  so  that  our  school  children  have  been  con- 
spicuously wanting  in  the  power  of  rapid  and  accurate  fig- 
uring.2 

In  this  connection  we  consider  the  approximations  to  the 
value   of  TT.     The    early   European    computers   followed    the 

1  GRAVES,  Life  of  Sir  Wm.  Eowan  Hamilton,  III.,  p.  275. 

2  The  following,  quoted  by  Mr.  E.  M.  Langley  in  the  Eighteenth  Gen- 
eral Report  of  the  A.  L  G.  T.,  1892,  p.  40,  from  De  Morgan's  article  "On 
Arithmetical  Computation"  in  the  Companion  to  the  British  Almanac 
for  1844,  is  interesting:  "The  growth  of  the  power  of  computation  on 
the  Continent,  though  considerable,  did  not  keep  pace  with  that  of  the 
same  in  England.     We  might  give  many  instances  of  the  truth  of  our 
assertion.     In  1696  De  Lagny,  a  well-known  writer  on  algebra,  and  a 
member  of  the  Academy  of  Sciences,  said  that  the  most  skilful  computer 
could  not,  in  less  than  a  month,  find  within  a  unit  the  cube  root  of 
696536483318640035073641037.    We  would  have  given  something  to  have 
been  present  if  De  Lagny  had  ever  made  the  assertion  to  his  contempo- 
rary, Abraham  Sharp.    In  the  present  day,  however,  both  in  our  uni- 
versities at  homo  and  everywhere  abroad,  no  disposition  to  encourage 
computation  exists  among  those  who  attend  to  the  higher  branches  of 
mathematics,  and  the  elementary  works  are  very  deficient  in  numerical 
examples."     De  Lagny 's  example  was  brought  to  De  Morgan  in  a  class, 
and  he  found  the  root  to  five  decimals  in  less  than  twenty  minutes.    Mr. 
Langley  exhibits  De  Morgan's  computation  on  p.  41  of  the  article  cited. 
Mr.  Langley  and  Mr.  R.  B.  Hayward  advocate  Horner's  method  as  a 
desirable  substitute  for  "  the  clumsy  rules  for  evolution  which  the  young 
student  still  usually  encounters  in  the  text-books."     See  HAYWARD'S 
article  in  the  A.  L  G.  T.  Report,  1889,  pp.  59-68,  also  DE  MORGAN'S 
article,  "Involution,"  in  the  Penny  or  the  English  Cyclopedia. 


242  A   HISTORY   OF  MATHEMATICS 

geometrical  method  of  Archimedes  by  inscribed  or  circum- 
scribed polygons.  Thus  Vieta,  about  1580,  computed  to  ten 
places,  Adrianus  Romanus  (1561-1615),  of  Louvain,  to  15 
places,  Ludolph  van  Ceulen  (1540-1610)  to  35  places.  The 
latter  spent  years  in  this  computation,  and  his  performance 
was  considered  so  extraordinary  that  the  numbers  were  cut 
on  his  tombstone  in  St.  Peter's  churchyard  at  Leyden.  The 
tombstone  is  lost,  but  a  description  of  it  is  extant.  After  him, 
the  value  of  IT  is  often  called  "  Ludolph's  number."  In  the 
seventeenth  century  it  was  perceived  that  the  computations 
could  be  greatly  simplified  by  the  use  of  infinite  series.  Such 

a?     x1     x7 

a   series,   viz.   tan"1  x  =  x 1 !-•••?   was    first    sug- 

357 

gested  by  James  Gregory  in  1671.  Perhaps  the  easiest  are  the 
formulae  used  by  Machin  and  Base.  Machin's  formula  is, 


The  Englishman,  Abraham  Sharp,  a  skilful  mechanic  and 
computor,  for  a  time  assistant  to  the  astronomer  Flamsteed, 
took  the  arc  in  Gregory's  formula  equal  to  30°,  and  calculated 
TT  to  72  places  in  1705 ;  next  year  Machin,  professor  of  astron- 
omy in  London,  gave  TT  to  100  places;  the  Frenchman,  De 
Lagny,  about  1719,  gave  127  places ;  the  German,  Georg  Vesa, 
in  1793,  140  places ;  the  English,  Rutherford,  in  1841,  208 
places  (152  correct) ;  the  German,  Zacharias  Dase,  in  1844, 
205  places ;  the  German,  Th.  Clausen,  in  1847,  250  places ;  the 
English,  Eutherf  ord,  in  1853,  440  places ;  William  Shanks,  in 
1873,  707  places.1  It  may  be  remarked  that  these  long  com- 
putations are  of  no  theoretical  or  practical  value.  Infinitely 
more  interesting  and  useful  are  Lambert's  proof  of  1761 

1  W.  W.  R.  BALL,  Math.  Recreations  and  Problems,  pp.  171-173. 
Ball  gives  bibliographical  references. 


THE   LAST   THREE   CENTURIES  243 

that  TT  is  not  rational,1  and  Lindemann's  proof  that  TT  is  not 
algebraical,  i.e.  cannot  be  the  root  of  an  algebraic  equation. 

Infinite  series  by  which  TT  may  be  computed  were  given  also 
by  Hutton  and  Euler.  Leonhard  Euler  (1707-1783),  of  Basel, 
contributed  vastly  to  the  progress  of  higher  mathematics,  but 
his  influence  reached  down  to  elementary  subjects.  He  treated 
trigonometry  as  a  branch  of  analysis,  introduced  (simultane- 
ously with  Thomas  Simpson  in  England)  the  now  current 
abbreviations  for  trigonometric  functions,  and  simplified  for- 
mulae by  the  simple  expedient  of  designating  the  angles  of 
a  triangle  by  A,  B,  G,  and  the  opposite  sides  by  a,  6,  c.  In 
his  old  age,  after  he  had  become  blind,  he  dictated  to  his 
servant  his  Anleitung  zur  Algebra,  1770,  which,  though  purely 
elementary,  is  meritorious  as  one  of  the  earliest  attempts  to  put 
the  fundamental  processes  upon  a  sound  basis.  An  Introduction 
to  the  Elements  of  Algebra,  .  .  .  selected  from  the  Algebra  of  Euler, 
was  brought  out  in  1818  by  John  Farrar  of  Harvard  College. 

A  question  that  became  prominent  toward  the  close  of  the 
eighteenth  century  was  the  graphical  representation  and  inter- 
pretation of  the  imaginary,  V— 1.  As  with  negative  numbers, 
so  with  iniaginaries,  no  decided  progress  was  made  until  a 
picture  of  it  was  presented  to  the  eye.  In  the  time  of  Newton, 
Descartes,  and  Euler,  the  imaginary  was  still  an  algebraic 
fiction.  The  first  thoroughly  successful  graphic  representations 
were  described  by  the  Norwegian  surveyor,  Caspar  Wessel,  in  an 
article  of  1797,  published  in  1799  in  the  Memoirs  of  the  Danish 
E.  Academy  of  Sciences,  and  by  Jean  Robert  Argand  (1768-?), 
of  Geneva,  who  in  1806  published  a  remarkable  Essai.2  But 

1  See  the  proof  in  Note  IV.  of  Legendre's  Geomttrie,  where  it  is  ex- 
tended to  ir2. 

2  Consult  Imaginary  Quantities.      Their  Geometrical  Interpretation. 
Translated  from  the  French  of  M.   ARGAND  by   A.   S.  HARDY, 
York,  1881. 


244  A    HISTORY   OF   MATHEMATICS 

all  these  writings  were  little  noticed,  and  it  remained  for  the 
great  Carl  Friedrich  Gauss  (1777-1855),  of  Gottingen,  to  break 
down  the  last  opposition  to  the  imaginary.  He  introduced 
it  as  an  independent  unit  co-ordinate  to  1,  and  a  +  ib  as  a 
"complex  number."  Notwithstanding  the  acceptance  of  imagi- 
naries  as  "  numbers "  by  all  great  investigators  of  the  nine- 
teenth century,  there  are  still  text-books  which  represent  the 
obsolete  view  that  V  —  1  is  not  a  number  or  is  not  a  quantity. 
Clear  ideas  on  the  fundamental  principles  of  algebra  were 
not  secured  before  the  nineteenth  century.  As  late  as  the 
latter  part  of  the  eighteenth  century  we  find  at  Cambridge, 
England,  opposition  to  the  use  of  the  negative.1  The  view 
was  held  that  there  exists  no  distinction  between  arithmetic 
and  algebra.  In  fact,  such  writers  as  Maclaurin,  Saunderson, 
Thomas  Simpson,  Hutton,  Bonnycastle,  Bridge,  began  their 
treatises  with  arithmetical  algebra,  but  gradually  and  dis- 
guisedly  introduced  negative  quantities.  Early  American 
writers  imitated  the  English.  But  in  the  nineteenth  century 
the  first  principles  of  algebra  came  to  be  carefully  investi- 
gated by  George  Peacock,2  D.  F.  Gregory,3  De  Morgan.4  Of 
continental  writers  we  may  mention  Augustin  Louis  Cauchy 
(1789-1857),5  Martin  Ohm,6  and  especially  Hermann  Hankel.7 

1  See  C.WORDSWORTH,  Scholce  Academicce :  Some  Account  of  the  Studies 
at  English  Universities  in  the  Eighteenth  Century,  1877,  p.  68 ;   Teach, 
and  Hist,  of  Math,  in  the  U.  S.,  pp.  385-387. 

2  See  his  Algebra,  1830  and  1842,  and  his  "  Report  on  Recent  Progress 
in  Analysis,"  printed  in  the  Reports  of  the  British  Association,  1833. 

3  "On  the  Real  Nature  of  Symbolical  Algebra,"   Trans.  Hoy.  Soc. 
Edinburgh,  Vol.  XIV.,  1840,  p.  280. 

4  "On  the  Foundation  of  Algebra,"   Cambridge  Phil   Trans.,  VII., 
1841,  1842  ;  VIII.,  1844,  1847. 

5  Analyse  Algebrique,  1821,  p.  173  et  seq. 

6  Versuchs  eines  vollkommen  consequenten  Systems  der  Mathematik, 
1822,  2d  Ed.  1828. 

7  Die  Complexen  Zahlen,  Leipzig,  1867.      This  work  is  very  rich  in 
historical  notes.    Most  of  the  bibliographical  references  on  this  subject 
given  here  are  taken  from  that  work. 


EDITIONS    OF   EUCLID  245 

A  flood  of  additional  light  has  been  thrown  on  this  subject 
by  the  epoch-making  researches  of  William  Rowan  Hamilton, 
Hermann  Grassmann,  and  Benjamin  Peirce,  who  conceived 
new  algebras  with  laws  differing  from  the  laws  of  ordinary 
algebra.1 

GEOMETRY   AND   TRIGONOMETRY 

Editions  of  Euclid.     Early  Researches 

With  the  close  of  the  fifteenth  century  and  beginning  of 
the  sixteenth  we  enter  upon  a  new  era.  Great  progress  was 
made  in  arithmetic,  algebra,  and  trigonometry,  but  less  prom- 
inent were  the  advances  in  geometry.  Through  the  study  of 
Greek  manuscripts  which,  after  the  fall  of  Constantinople  in 
1453,  came  into  possession  of  Western  Europe,  improved  trans- 
lations of  Euclid  were  secured.  At  the  beginning  of  this 
period,  printing  was  invented ;  books  became  cheap  and  plenti- 
ful. The  first  printed  edition  of  Euclid  was  published  in 
Venice,  1482.  This  was  the  translation  from  the  Arabic  by 
Campanus.  Other  editions  of  this  appeared,  at  Ulm  in  1486, 
at  Basel  in  1491.  The  first  Latin  edition,  translated  from 
the  original  Greek,  by  Bartholomceus  Zambertus,  appeared  at 
Venice  in  1505.  In  it  the  translation  of  Campanus  is  severely 
criticised.  This  led  Pacioli,  in  1509,  to  bring  out  an  edition, 
the  tacit  aim  of  which  seems  to  have  been  to  exonerate  Cam- 
panus.2 Another  Euclid  edition  appeared  in  Paris,  1516.  The 
first  edition  of  Euclid  printed  in  Greek  was  brought  out  in 
Basel  in  1533,  edited  by  Simon  Grynceus.  For  170  years  this 
was  the  only  Greek  text.  In  1703  David  Gregory  brought 
out  at  Oxford  all  the  extant  works  of  Euclid  in  the  original. 

1  For  an  excellent  historical  sketch  on  Multiple  Algebra,  see  J.  W. 
GIBBS,  in  Proceed.  Am.  Ass.  for  the  Adv.  of  Science,  Vol.  XXXV.,  1886. 

2  CANTOR,  II.,  p.  312. 


246  A   HISTORY  OF  MATHEMATICS 

As  a  complete  edition  of  Euclid,  this  stood  alone  until  1883, 
when  Heiberg  and  Menge  began  the  publication,  in  Greek 
and  Latin,  of  their  edition  of  Euclid's  works.  The  first 
English  translation  of  the  Elements  was  made  in  1570  from 
the  Greek  by  "H.  Billingsley,  Citizen  of  London."1  An 
English  edition  of  the  Elements  and  the  Data  was  published 
in  1758  by  Robert  Simson  (1687-1768),  professor  of  mathe- 
matics at  the  University  of  Glasgow.  His  text  was  until 
recently  the  foundation  of  nearly  all  school  editions.  It  dif- 
fers considerably  from  the  original.  Simson  corrected  a  num- 
ber of  errors  in  the  Greek  copies.  All  these  errors  he  assumed 
to  be  due  to  unskilful  editors,  none  to  Euclid  himself.  A  close 
English  translation  of  the  Greek  text  was  made  by  James 
Williamson.  The  first  volume  appeared  at  Oxford  in  1781 ; 
the  second  volume  in  1788.  School  editions  of  the  Elements 
usually  contain  the  first  six  books,  together  with  the  eleventh 
and  twelfth. 

Returning  to  the  time  of  the  Renaissance,  we  mention  a  few 

1  In  the  General  Dictionary  by  BAYLE,  London,  1735,  it  says  that 
Billingsley  "made  great  progress  in  mathematics,  by  the  assistance  of 
his  friend,  Mr.  Whitehead,  who,  being  left  destitute  upon  the  dissolution 
of  the  monasteries  in  the  reign  of  Henry  VIII.,  was  received  by  Billings- 
ley into  his  family,  and  maintained  by  him  in  his  old  age  in  his  house  at 
London."  Billingsley  was  rich  and  was  Lord  Mayor  of  London  in  1591. 
Like  other  scholars  of  his  day,  he  confounded  our  Euclid  with  Euclid  of 
Megara.  The  preface  to  the  English  edition  was  written  by  John  Dee, 
a  famous  astrologer  and  mathematician.  An  interesting  account  of  Dee 
is  given  in  the  Dictionary  of  National  Biography.  De  Morgan  thought 
that  Dee  had  made  the  entire  translation,  but  this  is  denied  in  the  article 
"Billingsley"  of  this  dictionary.  At  one  time  it  was  believed  that  Bil- 
lingsley translated  from  an  Arabic-Latin  version,  but  G.  B.  Halsted  suc- 
ceeded in  proving  from  a  folio —  once  the  property  of  Billingsley  —  [now 
in  the  library  of  Princeton  College,  and  containing  the  Greek  edition  of 
1533,  together  with  some  other  editions]  that  Billingsley  translated  from 
the  Greek,  not  the  Latin.  See  "Note  on  the  First  English  Euclid"  in 
the  Am.  Jour,  of  Mathem.,  Vol.  II.,  1879. 


EARLY    RESEARCHES  247 

of  the  more  interesting  problems  then  discussed  in  geometry. 
Of  a  development  of  new  geometrical  methods  of  investigation 
we  find  as  yet  no  trace.  In  his  book,  De  triangulis,  1533,  the 
German  astronomer,  Regiomontanus,  gives  the  theorem  (already 
known  to  Proclus)  that  the  three  perpendiculars  from  the  ver- 
tices of  a  triangle  meet  in  a  point ;  and  shows  how  to  find  from 
the  three  sides  the  radius  of  the  circumscribed  circle.  He  gives 
the  first  new  maximum  problem  considered  since  the  time  of 
Apollonius  and  Zenodorus,  viz.,  to  find  the  point  on  the  floor 
(or  rather  the  locus  of  that  point)  from  which  a  vertical  10 
foot  rod,  whose  lower  end  is  4  feet  above  the  floor,  seems 
largest  (i.e.  subtends  the  largest  angle).1  ISTew  is  the  following 
thfbrem,  which  brings  out  in  bold  relief  a  fundamental  differ- 
ence between  the  geometry  on  a  plane  and  the  geometry  on  a 
sphere :  From  the  three  angles  of  a  spherical  triangle  may  be 
computed  the  three  sides,  and  vice  versa.  Regiomontanus  dis- 
cussed also  star-polygons.  He  was  probably  familiar  with  the 
writings  on  this  subject  of  Campanus  and  Bradwardine.  Regi- 
omontanus, and  especially  the  Frenchman,  Charles  de  Bouvelles, 
or  Carolus  Bovillus  (1470-1533),  laid  the  foundation  for  the 
theory  of  regular  star-polygons.2 

The  construction  of  regular  inscribed  polygons  received  the 

1  CANTOR,  II.,  283. 

2  A  detailed  history  of  star-polygons  and  star-polyaedra  is  given  by 
S.  GUNTHER,  Vermischte  Untersuchungen,  pp.  1-92.     Star-polygons  have 
commanded   the    attention   of   geometers  down  even   to  recent  times. 
Among  the   more  prominent  are   Petrus    Ramus,   Athanasius   Kircher 
(1602-1680),  Albert  Girard,  John  Broscius  (a  Pole),  John  Kepler,  A.  L. 
F.  Meister  (1724-1788),  C.  F.  Gauss,  A.  F.  Mobius,  L.  Poinsot  (1777- 
1859),  C.  C.  Krause.     Mobius  gives  the  following  definition  for  the  area 
of  a  polygon,  useful  in  case  the  sides  cross  each  other :  Given  an  arbi- 
trarily formed  plane  polygon  AB  .  .  .  MN;  assume  any  point  P  in  the 
plane  and  connect  it  with  the  vertices  by  straight  lines;  then  the  sum 
PAB  +  PBC  +  .  .  .  +  PMN  +  PNA  is  independent  of  the  position  of 
P  and  represents  the  area  of  the  polygon.    Here  PAB  =  —  PBA. 


248  A   HISTORY   OP   MATHEMATICS 

attention  of  the  great  painter  and  architect,  Leonardo  da  Vinci 
(1452-1519).  Of  his  methods  some  are  mere  approximations, 
of  no  theoretical  interest,  though  not  without  practical  value. 
His  inscription  of  a  regular  heptagon  (of  course,  merely  an 
approximation)  he  considered  to  be  accurate !  Similar  construc- 
tions were  given  by  the  great  German  painter,  Albrecht  Diirer, 
(1471-1528).  He  is  the  first  who  always  clearly  and  correctly 
states  which  of  the  constructions  are  approximations.1  Both 
Leonardo  da  Vinci  and  Diirer  in  some  cases  perform  a  construc- 
tion by  using  one  single  opening  of  the  compasses.  Pappus 
once  set  himself  this  limitation ;  Abul  Wafa  did  this  repeat- 
edly ;  but  now  this  method  becomes  famous.  It  was  used  by 
Tartaglia  in  67  different  constructions ;  it  was  employed  2*so 
by  his  pupil  Giovanni  Battista  Benedetti  (1530-1590). 2 

It  will  be  remembered  that  Greek  geometers  demanded  that 
all  geometric  constructions  be  effected  by  a  ruler  and  compasses 
only ;  other  methods,  which  have  been  proposed  from  time  to 
time,  are  to  construct  by  the  compasses  only  or  by  the  ruler 
only,3  or  by  ruler,  compasses,  and  other  additional  instruments. 
Constructions  of  the  last  class  were  given  by  the  Greeks,  but 
were  considered  by  them  mechanical,  not  geometric.  A  peculiar 
feature  in  the  theory  of  all  these  methods  is  that  elementary 

1  CANTOR,  II.,  462. 

2  For  further  details,  consult  CANTOR  II.,  296-300,    450,  527,    529; 
S.  GUNTHER,  Nachtrdge,  p.  117,  etc.    The  fullest  development  of  this 
pretty  method  is  reached  in  STEINER,  Die  Geometrischen  Constructionen, 
ausgefuhrt  mittels  der  geraden  Linie  and  eines  festen  Kreises,  Berlin, 
1833 ;  and  in  PONCELET,   Traite  des  proprietes  projectives,  Paris,  1822, 
p.  187,  etc. 

8  Problems  to  be  solved  by  aid  of  the  ruler  only  are  given  by  LAMBERT 
in  his  Freie  Perspective,  Zurich,  1774  ;  by  SERVOIS,  Solutions  pen  connues 
de  differens  problemes  de  Geometric  pratique,  1805  ;  BRIANCHON,  Memoire 
sur  V 'application  de  la  theorie  des  transver sales.  See  also  CHASLES, 
p.  210;  CREMONA,  Elements  of  Protective  Geometry,  Transl.  by  LEUDES- 
DORF,  Oxford,  1885,  pp.  xiL,  96-98. 


EARLY   RESEARCHES  249 

geometry  is  unable  to  answer  the  general  question,  What  con- 
structions can  be  carried  out  by  either  one  of  these  methods  ? 
For  an  answer  we  must  resort  to  algebraic  analysis.1 

A  construction  by  other  instruments  than  merely  the  ruler 
and  compasses  appears  in  the  quadrature  of  the  circle  by  Leo- 
nardo da  Vinci.  He  takes  a  cylinder  whose  height  equals  half 
its  radius ;  its  trace  on  a  plane,  resulting  from  one  revolution, 
is  a  rectangle  whose  area  is  equal  to  that  of  the  circle.  Nothing 
could  be  simpler  than  this  quadrature;  only  it  must  not  be 
claimed  that  this  solves  the  problem  as  the  Greeks  understood 
it.  The  ancients  did  not  admit  the  use  of  a  solid  cylinder  as  an 
instrument  of  construction,  and  for  good  reasons :  while  with  a 
ruler  we  can  easily  draw  a  line  of  any  length,  and,  with  an  or- 
dinary pair  of  compasses,  any  circle  needed  in  a  drawing,  we 
can  with  a  given  cylinder  effect  not  a  single  construction  of 
practical  value.  No  draughtsman  ever  thinks  of  using  a 
cylinder.2 

To  Albrecht  Diirer  belongs  the  honour  of  having  shown  how 
the  regular  and  the  semi-regular  solids  can  be  constructed  out 
of  paper  by  marking  off  the  bounding  polygons,  all  in  one 
piece,  and  then  folding  along  the  connected  edges.3 

Polyedra  were  a  favourite  study  with  John  Kepler.  In 
1596,  at  the  beginning  of  his  extraordinary  scientific  career,  he 
made  a  pseudo-discovery  which  brought  him  much  fame.  He 
placed  the  icosaedron,  dodecaedron,  octaedron,  tetraedron,  and 
cube,  one  within  the  other,  at  such  distances  that  each  polye- 
dron  was  inscribed  in  the  same  sphere,  about  which  the  next 
outer  one  was  circumscribed.  On  imagining  the  sun  placed  in 
the  centre  and  the  planets  moving  along  great  circles  on  the 
spheres  —  taking  the  radius  of  the  sphere  between  the  icosae- 

1  KLEIN,  p.  2. 

2  For  a  good  article  on  the  "  Squaring  of  the  Circle,"  see  HERMANN 
SCHUBERT  in  Monist,  Jan.,  1891.  3  CANTOR,  II.,  466. 


250  A    HISTORY   OF    MATHEMATICS 

dron  and  dodecaedron  equal  to  the  radius  of  the  earth's  orbit 
—  he  found  the  distances  between  these  planets  to  agree  roughly 
with  astronomical  observations.  This  reminds  us  of  Pythag- 
orean mysticism.  But  maturer  reflection  and  intercourse 
with  Tycho  Brahe  and  Galileo  led  him  to  investigations  and 
results  more  worthy  of  his  genius  —  "  Kepler's  laws."  Kepler 
greatly  advanced  the  theory  of  s tar-poly  edra.1  An  innova- 
tion in  the  mode  of  geometrical  proofs,  which  has  since 
been  widely  used  by  European  and  American  writers  of 
elementary  books,  was  introduced  by  the  Frenchman,  Francis- 
cus  Vieta.  He  considered  the  circle  to  be  a  polygon  with 
an  infinite  number  of  sides.2  The  same  view  was  taken  by 
Kepler.  In  recent  times  this  geometrical  fiction  has  been 
generally  abandoned  in  elementary  works;  a  circle  is  not  a 
polygon,  but  the  limit  of  a  polygon.  To  advanced  mathema- 
ticians Vieta's  idea  is  of  great  service  in  simplifying  proofs, 
and  by  them  it  may  be  safely  used. 

The  revival  of  trigonometry  in  Germany  is  chiefly  due  to 
John  Mutter,  more  generally  called  Regiomontanus  (1436-1476). 
At  Vienna  he  studied  under  the  celebrated  Georg  Purbach, 
who  began  a  translation,  from  the  Greek,  of  the  Almagest, 
which  was  completed  by  Regiomontanus.  The  latter  also 
translated  from  the  Greek  works  of  Apollonius,  Archimedes, 
and  Heron.  Instead  of  dividing  the  radius  into  3438  parts, 
in  Hindu  fashion,  Regiomontanus  divided  it  into  600,000 
equal  parts,  and  then  constructed  a  more  accurate  table  of 
sines.  Later  he  divided  the  radius  into  10,000,000  parts. 
The  tangent  had  been  known  in  Europe  before  this  to  the 
Englishman,  Bradwardine,  but  Regiomontanus  went  a  step 
further,  and  calculated  a  table  of  tangents.  He  published  a 

1  For  drawings  of  Kepler's  star-polyedra  and  a  detailed  history  of  the 
subject,  see  S.  GCNTHER,  Verm.  Untersuchungen,  pp.  36-92. 

2  CANTOR,  II.,  586. 


EARLY   RESEARCHES  251 

treatise  on  trigonometry,  containing  solutions  of  plane  and 
spherical  triangles.  The  form  which  he  gave  to  trigonometry 
has  been  retained,  in  its  main  features,  to  the  present  day. 
The  task  of  computing  accurate  tables  was  continued  by 
the  successors  of  Regiomontanus.  More  refined  astronomical 
instruments  furnished  observations  of  greater  precision  and 
necessitated  the  computation  of  more  extended  tables  of 
trigonometric  functions.  Of  the  several  tables  calculated, 
that  of  Georg  Joachim  of  Feldkirch  in  Tyrol,  generally  called 
Rhceticus,  deserves  special  mention.  In  one  of  his  sine-tables, 
he  took  the  radius  =  1,000,000,000,000,000  and  proceeded  from 
10"  to  10".  He  began  also  the  construction  of  tables  of 
tangents  and  secants.  For  twelve  years  he  had  in  continual 
employment  several  calculators.  The  work  was  completed  by 
his  pupil,  Valentin  Otho,  in  1596.  A  republication  was  made 
by  Pitiscus  in  1613.  These  tables  are  a  gigantic  monument 
of  German  diligence  and  perseverance.  But  Rhseticus  was 
not  a  mere  computer.  Up  to  his  time  the  trigonometric 
functions  had  been  considered  always  with  relation  to  the 
arc;  he  was  the  first  to  construct  the  right  triangle  and  to 
make  them  depend  directly  upon  its  angles.  It  was  from  the 
right  triangle  that  he  took  his  idea  of  calculating  the  hypot- 
enuse, i.e.  the  secant.  His  was  the  first  European  table  of 
secants.  Good  work  in  trigonometry  was  done  also  by  Vieta, 
Adrianus  Eomanus,  Nathaniel  Torporley,  John  Napier,  Wille- 
brord  Snellius,  Pothenot,  and  others.  An  important  geodetic 
problem  —  given  a  terrestrial  triangle  and  the  angles  subtended 
by  the  sides  of  the  triangle  at  a  point  in  the  same  plane,  to 
find  the  distances  of  the  point  from  the  vertices  —  was  solved 
by  Snellius  in  a  work  of  1617,  and  again  by  Pothenot  in  1730. 
Snellius's  investigation  was  forgotten  and  it  secured  the  name 
of  "  Pothenot' s  problem."  The  word  "radian"  is  due  to 
James  Thomson,  1871. 


252  A   HISTORY   OF   MATHEMATICS 

The  Beginning  of  Modern  Synthetic  Geometry 

About  the  beginning  of  the  seventeenth  century  the  first 
decided  advance,  since  the  time  of  the  ancient  Greeks,  was 
made  in  Geometry.  Two  lines  of  progress  are  noticeable : 
(1)  the  analytic  path,  marked  out  by  the  genius  of  Descartes, 
the  inventor  of  Analytical  Geometry ;  (2)  the  synthetic  path, 
with  the  new  principle  of  perspective  and  the  theory  of  trans- 
versals. The  early  investigators  in  modern  synthetic  geome- 
try are  Desargues,  Pascal,  and  De  Lahire. 

Girard  Desargues  (1593-1662),  of  Lyons,  was  an  architect 
and  engineer.  Under  Cardinal  Richelieu  he  served  in  the 
siege  of  La  Eochelle,  in  1628.  Soon  after,  he  retired  to  Paris, 
where  he  made  his  researches  in  geometry.  Esteemed  by  the 
ablest  of  his  contemporaries,  bitterly  attacked  by  others  unable 
to  appreciate  his  genius,  his  works  were  neglected  and  forgot- 
ten, and  his  name  fell  into  oblivion  until,  in  the  early  part  of 
the  nineteenth  century,  it  was  rescued  by  Brianchon  and 
Poncelet.  Desargues,  like  Kepler  and  others,  introduced  the 
doctrine  of  infinity  into  geometry.1  He  states  that  the  straight 
line  may  be  regarded  as  a  circle  whose  centre  is  at  infinity ; 
hence,  the  two  extremities  of  a  straight  line  may  be  considered 
as  meeting  at  infinity;  parallels  differ  from  other  pairs  of 
lines  only  in  having  their  points  of  intersection  at  infinity. 
He  gives  the  theory  of  involution  of  six  points,  but  his 
definition  of  "  involution  "  is  not  quite  the  same  as  the  modern 
definition,  first  found  in  Fermat,2  but  really  introduced  into 
geometry  by  Chasles.3  On  a  line  take  the  point  A  as  origin 
(souche),  take  also  the  three  pairs  of  points  B  and  H,  C  and 
G,  D  and  F-,  then,  says  Desargues,  if  AB  -  AH—  AC-  A G 

1  CHARLES  TAYLOR,  Introduction  to  the  Ancient  and  Modern  Geometry 
of  Conies,  Cambridge,  1881,  p.  Ixi.  2  CANTOR,  II.,  677,  678. 

3  Consult  CHASLES,  Note  X. ;   MARIE,  III.,  214. 


SYNTHETIC    GEOMETRY  258 

=  AD  •  AF,  the  six  points  are  in  "  involution."  If  a  point  falls 
on  the  origin,  then  its  partner  must  be  at  an  infinite  distance 
from  the  origin.  If  from  any  point  P  lines  be  drawn  through 
the  six  points,  these  lines  cut  any  transversal  MN  in  six  other 
points,  which  are  also  in  involution ;  that  is,  involution  is  a 
projective  relation.  Desargues  also  gives  the  theory  of  polar 
lines.  What  is  called  "Desargues'  Theorem"  in  elementary 
works  is  as  follows :  If  the  vertices  of  two  triangles,  situated 
either  in  space  or  in  a  plane,  lie  on  three  lines  meeting  in  a 


point,  then  their  sides  meet  in  three  points  lying  on  a  line,  and 
conversely.  This  theorem  has  been  used  since  by  Brianchon, 
Sturm,  Gergonne,  and  others.  Poncelet  made  it  the  basis  of 
his  beautiful  theory  of  homological  figures. 

Although  the  papers  of  Desargues  fell  into  neglect,  his  ideas 
were  preserved  by  his  disciples,  Pascal  and  Philippe  de  Lahire. 
The  latter,  in  1679,  made  a  complete  copy  of  Desargues'  princi- 
pal research,  published  in  1639.  Blaise  Pascal  (1623-1662) 
was  one  of  the  very  few  contemporaries  who  appreciated  the 
worth  of  Desargues.  He  says  in  his  Essais  pour  les  coniques, 
"  I  wish  to  acknowledge  that  I  owe  the  little  that  I  have  dis- 
covered on  this  subject  to  his  writings."  Pascal's  genius  for 
geometry  showed  itself  when  he  was  but  twelve  years  old. 
His  father  wanted  him  to  learn  Latin  and  Greek  before  enter- 
ing on  mathematics.  All  mathematical  books  were  hidden 
out  of  sight.  In  answer  to  a  question,  the  boy  was  told  by 


254  A   HISTORY    OF   MATHEMATICS 

his  father  that  mathematics  "was  the  method  of  making 
figures  with  exactness,  and  of  finding  out  what  proportions 
they  relatively  had  to  one  another."  He  was  at  the  same  time 
forbidden  to  talk  any  more  about  it.  But  his  genius  could 
not  be  thus  confined;  meditating  on  the  above  definition,  he 
drew  figures  with  a  piece  of  charcoal  upon  the  tiles  of  the 
pavement.  He  gave  names  of  his  own  to  these  figures, 
then  formed  axioms,  and,  in  short,  came  to  make  perfect 
demonstrations.  In  this  way  he  arrived,  unaided,  at  the 
theorem  that  the  angle-sum  in  a  triangle  is  two  right  angles. 
His  father  caught  him  in  the  act  of  studying  this  theorem,  and 
was  so  astonished  at  the  sublimity  and  force  of  his  genius  as 
to  weep  for  joy.  The  father  now  gave  him  Euclid's  Elements, 
which  he  mastered  easily.  Such  is  the  story  of  Pascal's  early 
boyhood,  as  narrated  by  his  devoted  sister.1  While  this  narra- 
tive must  be  taken  cum  grano  salis  (for  it  is  highly  absurd  to 
suppose  that  young  Pascal  or  any  one  else  could  re-discover 
geometry  as  far  as  Euclid  I.,  32,  following  the  same  treatment 
and  hitting  upon  the  same  sequence  of  propositions  as  found 
in  the  Elements),  it  is  true  that  Pascal's  extraordinary  pene- 
tration enabled  him  at  the  age  of  sixteen  to  write  a  treatise  on 
conies  which  passed  for  such  a  surprising  effort  of  genius  that 
it  was  said  nothing  equal  to  it  in  power  had  been  produced 
since  the  time  of  Archimedes.  Descartes  refused  to  believe 
that  it  was*  written  by  one  so  young  as  Pascal.  This  treatise 
was  never  published,  and  is  now  lost.  Leibniz  saw  it  in  Paris, 
recommended  its  publication,  and  reported  on  a  portion  of  its 
contents.2  However,  Pascal  published  in  1640,  when  he  was 
sixteen  years  old,  a  small  geometric  treatise  of  six  octavo 

1  The  Life  of  Mr.  Paschal,  by  MADAM  PERIER.    Translated  into  Eng- 
lish by  W.  A.,  London,  1744. 

2  See  letter  written  by  Leibniz  to  Pascal's  nephew,  August  30,  1676, 
which  is  given  in  Oeuvres  completes  de  Blaise  Pascal,  Paris,  1866,  Vol.  III., 


SYNTHETIC    GEOMETRY  255 

pages,  bearing  the  title,  Essais  pour  les  coniques.  Constant 
application  at  a  tender  age  greatly  impaired  Pascal's  health. 
During  his  adult  life  he  gave  only  a  small  part  of  his  time  to 
the  study  of  mathematics. 

Pascal's  two  treatises  just  noted  contained  the  celebrated 
proposition  on  the  mystic  hexagon,  known  as  "Pascal's 
Theorem,"  viz.  that  the  opposite  sides  of  a  hexagon  inscribed 
in  a  conic  intersect  in  three  points  which  are  collinear.  In 
our  elementary  text-books  on  modern  geometry  this  beautiful 
theorem  is  given  in  connection  with  a  very  special  type  of  a 
conic,  namely,  the  circle.  As,  in  one  sense,  any  two  straight 
lines  may  be  looked  upon  as  a  special  case  of  a  conic,  the 
theorem  applies  to  hexagons  whose  first,  third,  and  fifth  ver- 
tices are  on  one  line,  and  whose  second,  fourth,  and  sixth  ver- 
tices are  on  the  other.  It  is  interesting  to  note  that  this 
special  case  of  "  Pascal's  Theorem  "  occurs  already  in  Pappus 
(Book  VII.,  Prop.  139).  Pascal  said  that  from  his  theorem 
he  deduced  over  400  corollaries,  embracing  the  conies  of 
Apollonius  and  many  other  results.  Pascal  gave  the  theorem 
on  the  cross  ratio,  first  found  in  Pappus.1  This  wonderfully 
fruitful  theorem  may  be  stated  as  follows  :  Four  lines  in  a 
plane,  passing  through  one  common  point,  cut  off  four  seg- 
ments on  a  transversal  which  have  a  fixed,  constant  ratio,  in 
whatever  manner  the  transversal  may  be  drawn;  that  is,  if 

the  transversal  cuts  the  rays  in  the  points  A,  B,  C,  D,  then  the 

AG     BC 

ratio  -  :  -  ,  formed  by  the  four  segments  AC,  AD,  BC, 

JjJLs 


BD,  is  the  same  for  all  transversals.  The  researches  of 
Desargues  and  Pascal  uncovered  several  of  the  rich  treasures 

pp.  466-468.  The  Essais  pour  les  coniques  is  given  in  Vol.  III.,  pp. 
182-185,  of  the  Oeuvres  completes,  also  in  Oeuvres  de  Pascal  (The  Hague, 
1779)  and  by  H.  WEISSENBORN  in  the  preface  to  his  book,  Die  Projection 
in  der  Ebene,  Berlin,  1862. 

1  Book  VII.,  129.     Consult  CHASLES,  pp.  31,  32. 


256  A  HISTORY  OF   MATHEMATICS 

of  modern  synthetic  geometry;  but  owing  to  the  absorbing 
interest  taken  in  the  analytical  geometry  of  Descartes,  and, 
later,  in  the  differential  calculus,  the  subject  was  almost  en- 
tirely neglected  until  the  close  of  the  eighteenth  century. 

Synthetic  geometry  was  advanced  in  England  by  the  re- 
searches of  Sir  Isaac  Newton,  Roger  Cotes  (1682-1716),  and 
Colin  Maclaurin,  but  their  investigations  do  not  come  within 
the  scope  of  this  history.  Robert  Simson  and  Matthew  Stewart 
(1717-1785)  exerted  themselves  mainly  to  revive  Greek  ge- 
ometry. An  Italian  geometer,  Giovanni  Ceva  (1647-1734) 1 
deserves  mention  here;  a  theorem  in  elementary  geometry 
bears  his  name.  He  was  an  hydraulic  engineer,  and  as  such 
was  several  times  employed  by  the  government  of  Mantua. 
His  death  took  place  during  the  siege  of  Mantua,  in  1734.  He 
ranks  as  a  remarkable  author  in  economics,  being  the  first 
clear-sighted  mathematical  writer  on  this  subject.  In  1678  he 
published  in  Milan  a  work,  De  lineis  rectis.  This  contains 
"  Ceva's  Theorem  "  with  one  static  and  two  geometric  proofs  : 

Any    three    concurrent   lines 
through  the  vertices  of  a  triangle 
divide  the  opposite  sides  so  that 
.B     Ca- Aj3>  By  =  Ba-  C/3  •  Ay.     In 
Ceva's  book  the  properties  of  rec- 
tilinear figures  are  proved  by  considering  the  properties  of 
the  centre  of  inertia  (gravity)  of  a  system  of  points.2 

Modern  Elementary  Geometry 

We  find  it  convenient  to  consider  this  subject  under  the 
following  four  sub-heads :  (1)  Modern  Synthetic  Geometry, 
(2)  Modern  Geometry  of  the  Triangle  and  Circle,  (3)  Non- 

1  Bulletin  Am.  Math.  Soc.,  Vol.  22,  1915,  p.  100. 
!»CHASLES,  Notes  VI.,  VII. 


SYNTHETIC   GEOMETRY  257 

Euclidean  Geometry,  (4)  Text-books  on  Elementary  Geometry. 
The  first  of  these  divisions  has  reference  to  modern  synthetic 
methods  of  research,  the  second  division  refers  to  new  theorems 
in  elementary  geometry,  the  third  considers  the  modern  con- 
ceptions of  space  and  the  several  geometries  resulting  there- 
from, the  fourth  discusses  questions  pertaining  to  geometrical 
teaching. 

I.  Modern  Synthetic  Geometry. — It  was  reserved  for  the 
genius  of  Gaspard  Monge  (1746-1818)  to  bring  modern  synthetic 
geometry  into  the  foreground,  and  to  open  up  new  avenues 
of  progress.  To  avoid  the  long  arithmetical  computations  in 
connection  with  plans  of  fortification,  this  gifted  engineer  sub- 
stituted geometric  methods  and  was  thus  led  to  the  creation  of 
descriptive  geometry  as  a  distinct  branch  of  science.  Monge 
was  professor  at  the  Normal  School  in  Paris  during  the  four 
months  of  its  existence,  in  1795 ;  he  then  became  connected 
with  the  newly  established  Polytechnic  School,  and  later 
accompanied  Napoleon  on  the  Egyptian  campaign.  Among 
the  pupils  of  Monge  were  Dupin,  Servois,  Brianchon,  Hachette, 
Biot,  and  Poncelet.  Charles  Julien  Brianchon,  born  in  Sevres 
in  1785,  deduced  the  theorem,  known  by  his  name,  from 
"Pascal's  Theorem"  by  means  of  Desargues'  properties  of 
what  are  now  called  polars.1  Brianchon's  theorem  says :  "  The 
hexagon  formed  by  any  six  tangents  to  a  conic  has  its  opposite 
vertices  connecting  concurrently.7'  The  point  of  meeting  is 
sometimes  called  the  "  Brianchon  point. " 

Lazare  Nicholas  Marguerite  Carnot  (1753-1823)  was  born  at 
Nolay  in  Burgundy.  At  the  breaking  out  of  the  Revolution 
he  threw  himself  into  politics,  and  when  coalesced  Europe,  in 
1793,  launched  against  France  a  million  soldiers,  the  gigantic 

1  Brianchon's  proof  appeared  in  "Memoirs  sur  les  Surfaces  courbes 
du  second  Degr6,"  in  Journal  de  VEcole  Poly  technique,  T.  VI.,  297-311, 
1806.  It  is  reproduced  by  TAYLOR,  op.  cit.,  p.  290. 


258  A    HISTORY   OF   MATHEMATICS 

task  of  organizing  fourteen  armies  to  meet  the  enemy  was 
achieved  by  him.  He  was  banished  in  1796  for  opposing 
Napoleon's  coup  d'etat.  His  Geometry  of  Position,  1803,  and 
his  Essay  on  Transversals,  1806,  are  important  contributions 
to  modern  plane  geometry.  By  his  effort  to  explain  the  mean- 
ing of  the  negative  sign  in  geometry  he  established  a  "  geome- 
try of  position  "  which,  however,  is  different  from  Von  Staudt' s 
work  of  the  same  name.  He  invented  a  class  of  general  theo- 
rems on  projective  properties  of  figures,  which  have  since  been 
studied  more  extensively  by  Poncelet,  Chasles,  and  others. 

Jean  Victor  Poncelet  (1788-1867),  a  native  of  Metz,  engaged 
in  the  Eussian  campaign,  was  abandoned  as  dead  on  the 
bloody  field  of  Krasnoi,  and  from  there  taken  as  prisoner  to 
Saratoff.  Deprived  of  books,  and  reduced  to  the  remem- 
brance of  what  he  had  learned  at  the  Lyceum  at  Metz  and 
the  Polytechnic  School,  he  began  to  study  mathematics  from 
its  elements.  Like  Bunyan,  he  produced  in  prison  a  famous 
work,  Traite  des  Proprietes  projectives  des  Figures,  first  pub- 
lished in  1822.  Here  he  uses  central  projection,  and  gives 
the  theory  of  "  reciprocal  polars."  To  him  we  owe  the  Law 
of  Duality  as  a  consequence  of  reciprocal  polars.  As  an  inde- 
pendent principle  it  is  due  to  Joseph  Diaz  Gergonne  (1771- 
1859).  We  can  here  do  no  more  than  mention  by  name  a  few 
of  the  more  recent  investigators :  Augustus  Ferdinand  Mdbius 
(1790-1868),  Jacob  Steiner  (1796-1863),  Michel  Chasles  (1793- 
1880),  Karl  Georg  Christian  von  Staudt  (1798-1867).  Chasles 
introduced  the  bad  term  anharmonic  ratio,  corresponding  to 
the  German  Doppelverhdltniss  and  to  Clifford's  more  desirable 
cross-ratio.  Von  Staudt  cut  loose  from  all  algebraic  formulae 
and  from  metrical  relations,  particularly  the  metrically  founded 
cross-ratio  of  Steiner  and  Chasles,  and  then  created  a  geom- 
etry of  position,  which  is  a  complete  science  in  itself,  inde- 
pendent of  all  measurement. 


ELEMENTARY   GEOMETRY 


259 


II.  Modern  Geometry  of  the  Triangle  and  Circle.  —  We  can- 
not give  a  full  history  of  this  subject,  but  we  hope  by  our 
remarks  to  interest  a  larger  circle  of  American  readers  in  the 
recently  developed  properties  of  the  triangle  and  circle.1  Fre- 
quently quoted  in  recent  elementary  geometries  is  the  "nine- 
point  circle."  In  the  triangle  ABC,  let  D,  E,  F  be  the  middle 
points  of  the  sides,  let  AL, 
BM,  ON  be  perpendicu- 
lars to  the  sides,  let  a,  b, 
c  be  the  middle  points  of 
AO,  BO,  CO,  then  a  cir- 
cle can  be  made  to  pass 
through  the  points,  L,  D, 
c,  E,  M,  a,  N,  F,  b;  this 
circle  is  the  "nine-point 

circle."  By  mistake,  the  earliest  discovery  of  this  circle 
has  been  attributed  to  Euler.2  There  are  several  indepen- 
dent discoverers.  In  England,  Benjamin  Sevan  proposed  in 
LeybouTn's  Mathematical  Repository,  I.,  18,  1804,  a  theorem 
for  proof  which  practically  gives  us  the  nine-point  circle. 


1  A  systematic  treatise  on  this  subject,  which  we  commend  to  students 
is  A.  EMMERICH'S  Die  Brocardschen  Gebilde,  Berlin,  1891.     Our  historical 
notes  are  taken  from  this  book  and  from  the  following  papers :  JULIUS 
LANGE,    Geschichte  des  Feuerbachschen   Kreises,   Berlin,    1894 ;    J.    S. 
MACKAY,  History  of  the  nine-point  circle,  pp.  19-57,  Early  history  of 
the  symmedian  point,   pp.    92-104,  in  the   Proceed,   of  the  Edinburgh 
Math.   Soc.,  Vol.  XL,  1892-93.      See  also  MACKAY,   The   Wallace  line 
and  the  Wallace  point  in  the  same  journal,  Vol.  IX.,  1891,  pp.  83-91 ; 
E.    LEMOINE'S    paper  in  Association  franqaise  pour  Vavancement   des 
Sciences,  Congres  de  Grenoble,  1885  ;  E.  VIGARIE,  in  the  same  publica- 
tion, Congres  de  Paris,  1889.     The  progress  in  the  geometry  of  the  tri- 
angle is  traced  by  VIGARIE  for  the  year  1890  in  Progreso  mat.  I. ,  101-106, 
128-134,  187-190 ;  for  the  year  1891  in  Journ.  de  Math,  elem.,  (4)  1. 
7-10,  34-36.     Consult  also  CASEY,  Sequel  to  Euclid. 

2  MACKAY,  op.  cit.,  Vol.  XI.,  p.  19. 


260  A   HISTORY   OF  MATHEMATICS 

The  proof  was  supplied  to  the  Repository,  Vol.  I.,  Part  1, 
p.  143,  by  John  Butterworth,  who  also  proposed  a  problem, 
solved  by  himself  and  John  Whitley,  from  the  general  tenor 
of  which  it  appears  that  they  knew  the  circle  in  question 
to  pass  through  all  nine  points.  These  nine  points  are  explic- 
itly mentioned  by  Brianchon  and  Poncelet  in  Gergonne's 
Annales  de  Mathtmatiques  of  1821.  In  1822,  Karl  Wilhelm 
Feuerbach  (1800-1834)  professor  at  the  gymnasium  in  Erlangen, 
published  a  pamphlet  in  which  he  arrives  at  the  nine-point 
circle,  and  proves  that  it  touches  the  incircle  and  the  ex- 
circles.  The  Germans  called  it  "  Feuerbach' s  Circle."  Many 
demonstrations  of  its  characteristic  properties  are  given  in  the 
article  above  referred  to.  The  last  independent  discoverer  of 
this  remarkable  circle,  so  far  as  known,  is  F.  S.  Davies,  in  an 
article  of  1827  in  the  Philosophical  Magazine,  II.,  29-31. 

In  1816  August  Leopold  Crelle  (1780-1855),  the  founder  of 
a  mathematical  journal  bearing  his  name,  published  in  Berlin 
a  paper  dealing  with  certain  properties  of  plane  triangles. 
He  showed  how  to  determine  a  point  O  inside  a  triangle,  so 
that  the  angles  (taken  in  the  same  order)  formed  by  the  lines 
joining  it  to  the  vertices  are  equal. 

In  the  adjoining  figure  the  three  marked  angles  are  equal. 
If  the  construction  be  made  so  as  to 
give  angle  fl'AC  =  tt'CB  =  Q'BA,  then 
a  second  point  Of  is  obtained.  The 
study  of  the  properties  of  these  new 
angles  and  new  points  led  Crelle  to 
exclaim :  "  It  is  indeed  wonderful  that 
so  simple  a  figure  as  the  triangle  is  so 
inexhaustible  in  properties.  How  many  as  yet  unknown 
properties  of  other  figures  may  there  not  be!"  Investiga- 
tions were  made  also  by  C.  F.  A.  Jacobi  of  Pforta  and  some 
of  his  pupils,  but  after  his  death,  in  1855,  the  whole  matter 


ELEMENTARY   GEOMETRY 


261 


was  forgotten.  In  1875  the  subject  was  again  brought  before 
the  mathematical  public  by  H.  Brocard,  who  had  taken  up 
this  study  independently  a  few  years  earlier.  The  work  of 
Brocard  was  soon  followed  up  by  a  large  number  of  investi- 
gators in  France,  England,  and  Germany.  The  new  researches 
gave  rise  to  an  extended  new  vocabulary  of  technical  terms. 
Unfortunately,  the  names  of  geometricians  which  have  been 
attached  to  certain  remarkable  points,  lines,  and  circles  are  not 
always  the  names  of  the  men  who  first  studied  their  properties. 
Thus,  we  speak  of  "Brocard  points"  and  "Brocard  angles,"  but 

C 


historical  research  brought  out  the  fact,  in  1884,  and  1886,  that 
these  were  the  points  and  lines  which  had  been  studied  by 
Crelle  and  C.  F.  A.  Jacobi.  The  "Brocard  Circle,"  is  Brocard's 
own  creation.  In  the  triangle  ABC,  let  O  and  O'  be  the  first 
and  second  "Brocard  point."  Let  A'  be  the  intersection  of 
Btt  and  CO';  B',  of  AM  and  CO;  C',  of  B&  and  An.  The 
circle  passing  through  A.'9  B',  C'  is  the  "  Brocard  circle." 
A'B'C'  is  "Brocard's  first  triangle."  Another  like  triangle, 
A"B"C"  is  called  "Brocard's  second  triangle."  The  points 
A",  B",  C"  together  with  O,  O',  and  two  other  points,  lie  in 
the  circumference  of  the  "Brocard  circle." 


262 


A   HISTORY    OF   MATHEMATICS 


In  1873  Emile  Lemoine  called  attention  to  a  particular  point 
within  a  plane  triangle  which  since  has  been  variously  called 
the  "Lemoine  point,"  "symmedian  point,"  and  "Grebe  point." 
Since  that  time  the  properties  of  this  point  and  of  the  lines 
and  circles  connected  with  it  have  been  diligently  investigated. 
To  lead  up  to  its  definition  we  premise  that,  in  the  adjoining 
figure,  if  CD  is  so  drawn  as  to  make  angles  a  and  b  equal,  then 

one  of  the  two  lines  AB  and  CD 
x  is  the  anti-parallel  of  the  other, 
D  with  reference  to  the  angle  O.1 
Now  OE,  the  bisector  of  AB, 
is  the  median  and  OF,  the  bisec- 
tor of  the  anti-parallel  of  AB, 
is  called  the  symmedian  (ab- 
breviated from  symetrique  de  la 
mediane}.  The  point  of  concurrence  of  the  three  symmedians 
in  a  triangle  is  called,  after  Tucker,  the  "  symmedian  point." 
Mackay  has  pointed  out  that  some  of  the  properties  of  this 
point,  recently  brought  to  light,  were  discovered  previously 
to  1873.  The  anti-parallels  of  a  triangle  which  pass  through 
its  symmedian  point,  meet  its  sides  in  six  points  which  lie 
on  a  circle,  called  the  "second  Lemoine  circle."  The  "first 
Lemoine  circle"  is  a  special  case  of  a  "Tucker  circle"  and 
concentric  with  the  "  Brocard  circle."  The  "  Tucker  circles  " 
maybe  thus  defined.  Let  DF' =  FE'  =  ED'-,  let,  moreover, 
the  following  pairs  of  lines  be  anti-parallels  to  each  other: 

1  The  definition  of  anti-parallels  is  attributed  by  EMMERICH  (p.  13,  note) 
to  Leibniz.  The  term  anti-parallel  is  defined  by  E.  STONE  in  his  New 
Mathem.  Diet.,  London,  1743.  Stone  gives  the  above  definition,  refers 
to  Leibniz  in  Acta  Erudit.,  1691,  p.  279,  and  attributes  to  Leibniz  a 
definition  different  from  the  above.  The  word  anti-parallel  is  given  in 
MURRAY'S  New  English  Dictionary  of  about  1660.  See  Jahrbuch  uber 
die  Fortschritte  der  Mathematik,  Bd.  XXII.,  1890,  p.  45  ;  Nature,  XLL, 
104-105. 


ELEMENTARY   GEOMETRY 


263 


D 


D' 


AB  and  ED',  BC  and  FE',  CA  and  DF' ;  then  the  six  points 
D,  D',  E,  E',  F,  F',  lie  on  a  "Tucker  circle."  Vary  the 
length  of  the  equal  anti-parallels, 
and  the  various  "Tucker  cir- 
cles" are  obtained.  Allied  to 
these  are  the  "Taylor  circles." 
Still  different  types  are  the 
"Neuberg  circles"  and  "Mac- 
kay  circles."  Perhaps  enough 
has  been  said  to  call  to  mind  the 
wonderful  advance  which  has 
been  made  in  the  geometry  of  B 
the  triangle  and  circle  during 
the  latter  half  of  the  nineteenth  century.  That  new  theorems 
should  have  been  found  in  recent  time  is  the  more  remark- 
able when  we  consider  that  these  figures  were  subjected  to 
close  examination  by  the  keen-minded  Greeks  and  the  long 

line    of    geometers    who 
p  have  since  appeared.1 

Of  interest  is  the  in- 
vention in  1853  by  Sarrut 
of  an  apparatus  for  gener- 
ating rectilinear  motions ; 
it  has  been  altogether 
overlooked.  Celebrated 
is  the  invention  in  1864 
of  an  apparatus  for  con- 
verting circular  into  recti- 
linear motion,2  by  A.  Peaucellier,  a  French  army  officer.  Let 

1  For  a  more  detailed  statement  of  researches  in  England  see  article 
"The  Recent  Geometry  of  the  Triangle,"  Fourteenth  General  Meport  of 
A.  I.  G.  T.,  1888,  pp.  35-46. 

2  PEAUCELLIER'S  articles  appeared  in  Nouvelles  Annales,  1804  and  1873. 


264  A    HISTORY   OF   MATHEMATICS 

ACBP  be  a  rhombus  whose  sides  are  less  than  the  equal  sides 
of  the  angle  AOB.  Imagine  the  straight  lines  which  are 
drawn  in  full  to  be  bars  or  "links,"  jointed  at  the  points  A,  B, 
C,  P,  Q,  0.  Make  C  describe  the  circle  and  P  will  describe 
'  the  straight  line  PD,  the  pivot  0  being  fixed.  Eemove  CQ, 
let  C  move  along  any  curve  in  the  plane,  then  P  will  trace  the 
inverse  of  that  curve  with  respect  to  0.  Peaucellier's  method 
of  linkages  was  developed  further  by  Sylvester.1 

Elsewhere  we  have  spoken  of  the  instruments  used  in 
geometrical  constructions  —  how  the  Greeks  used  the  ruler 
and  compasses,  and  how  later  a  ruler  with  a  fixed  opening 
of  the  compasses,  or  the  ruler  alone,  came  to  be  used  by  a 
few  geometers.  Of  interest  in  this  connection  is  a  work  by 
the  Italian  Lorenzo  Mascheroni  (1750-1800),  entitled  Geome- 
tria  del  compasso,  1797,  in  which  all  constructions  are  made 
with  a  pair  of  compasses,  but  without  restriction  to  a  fixed 
radius.  The  book  was  written  for  the  practical  mechanic, 
the  author  claiming  that  constructions  with  compasses  are 
more  accurate  than  those  with  a  ruler.2  The  work  secured  the 
attention  of  Napoleon  Bonaparte,  who  proposed  to  the  French 
mathematicians  the  following  problem :  To  divide  the  circum- 
ference of  a  circle  into  four  equal  parts  by  the  compasses  only. 
This  construction  is  as  follows :  Apply  the  radius  three  times 
to  the  circumference,  giving  the  arcs  AB,  BC,  CD.  Then  the 
distance  AD  is  a  diameter.  With  a  radius  equal  to  AC,  and 

1  SYLVESTER  in  Educ.   Times  Reprint,  Vol.  XXL,  58,  (1874).      See 
C.  TAYLOR,  Ancient  and  Modern  Geometry  of  Conies,  1881,  p.  LXXXVII. 
Consult  also  A.  B.  KEMPE,  How  to  draw  a  straight  line,  a  lecture  on 
linkages,  London,  1877. 

2  See   CHARLES   BUTTON'S   Philos.    and  Math.   Diet.,  London,   1815, 
article  "  Geometry  of  the  Compasses  "  ;  MARIE,  X.  p.  98  ;  KLEIN,  p.  26  ; 
STEINER,  Gesammelte  Werke,  L,  463;  Mascheroni' s  book  was  translated 
into  French  in  1798 ;  an  abridgment  of  it  by  HUTT  was  brought  out  in 
German  in  1880. 


ELEMENTARY   GEOMETRY  265 

with  centres  A  and  D,  draw  arcs  intersecting  at  E.  Then  EO, 
where  0  is  the  centre  of  the  given  circle,  is  the  chord  of  the 
quadrant  of  the  circle. 

The  inscription  of  the  regular  polygon  of  17  sides  was  first 
effected  by  Carl  Friedrich  Gauss  (1777-1855),  when  a  boy  of 
nineteen,  at  the  University  of  Gottingen,  March  30,  1796.  At 
that  time  he  was  undecided  whether  to  choose  ancient  lan- 
guages or  mathematics  as  his  specialty.  His  success  in  this 
inscription  led  to  the  decision  in  favour  of  mathematics.1 

A  curious  mode  of  construction  has  been  suggested  inde- 
pendently by  a  German  and  a  Hindu.  Constructions  are  to 
be  effected  by  the  folding  of  paper.  Hermann  Wiener,  in  1893, 
showed  how  to  construct,  by  folding,  the  nets  of  the  regular 
solids.  In  the  same  year,  Sundara  Row  published  a  little 
book  "  On  paper  folding  "  (Addison  &  Co.,  Madras),  in  which 
it  is  shown  how  to  construct  any  number  of  points  on  the 
ellipse,  cissoid,  etc.2 

In  connection  with  polyedra  the  theorem  that  the  number  of 
edges  falls  short  by  two  of  the  combined  number  of  vertices 
and  faces  is  interesting.  The  theorem  is  usually  ascribed  to 
Euler,  but  was  worked  out  earlier  by  Descartes.3  It  is  true 
only  for  polyedra  whose  faces  have  each  but  one  boundary ;  if 
a  cube  is  placed  upon  a  larger  cube,  then  the  upper  face  of  the 
larger  cube  has  two  boundaries,  an  inner  and  an  outer,  and  the 
theorem  is  not  true.  A  more  general  theorem  is  due  to 
F.  Lippich* 

1  Other  modes  of  inscription  of  the  17-sided  polygon  were  given  by 
VON  STAUDT,  in  Crelle,  24  (1842),  by  SCHROTER,  in  Crelle,  75  (1872). 
The  latter  uses  a  ruler  and  a  single  opening  of  the  compasses.  By  the 
compasses  only,  the  inscription  has  also  been  effected.  See  KLEIN, 
p.  27  ;  an  inscription  is  given  also  in  PAUL  BACHMANN,  Kreistheilung, 
Leipzig,  1872,  p.  67.  2  KLEIN,  p.  33. 

8  See  E.  DE  JONQUIERES  in  Biblioth.  Mathem.,  1890,  p.  43. 

*  LIPPICH,  "zur  Theorie  der  Polyeder"  Sitz.-Ber  d.  Wien.  Akad.,  Bd. 


266  A    HISTORY    OF   MATHEMATICS 

HI.  Non-Euclidean  Geometry.  —  The  history  of  this  subject 
centres  almost  wholly  in  the  theory  of  parallel  lines.  Prelimi- 
nary to  the  discussion  of  the  non-Euclidean  geometries,  it  may  be 
profitable  to  consider  the  various  efforts  towards  simplifying  and 
improving  the  parallel-theory  made,  (1)  by  giving  a  new  defini- 
tion of  parallel  lines,  or  by  assuming  a  new  postulate,  different 
from  Euclid's  parallel-postulate,  (2)  by  proving  the  parallel- 
postulate  from  the  nature  of  the  straight  line  and  plane  angle. 

Euclid's  definition,  parallel  straight  lines  are  such  as  are  in 
the  same  plane,  and  which  being  produced  ever  so  far  both  ways 
do  not  meet,  still  holds  its  place  as  the  best  definition  for  use  in 
elementary  geometry.  Anew  definition,  used  by  Posidonius 
(1st  Cent.  B.C.),  was  adopted  by  the  painter,  Albrecht  Durer.  He 
wrote  a  geometry,  first  printed  in  1525,  in  which  parallel  lines 
are  lines  everywhere  equally  distant.1  A  little  later  Clavius,  in 
his  edition  of  Euclid  of  1574,  in  a  remark,  assumes  that  a  line 
which  is  everywhere  equidistant  from  a  straight  line  is  it- 
self straight.  This  postulate  lies  hidden  in  the  definition  of 
Durer.  The  objection  to  this  definition  or  postulate  is  that 
it  is  an  advanced  theorem,  involving  the  difficult  consider- 
ation of  measurement,  embracing  the  whole  theory  of  incom- 
mensurables.  Moreover,  in  a  more  general  view  of  geometry 
it  must  be  abandoned.2  Clavius's  treatment  is  adopted  by 
Jacques  Peletier  (1517-1582)  of  Paris,  Ruggiero  Giuseppe 
Boscovich  (1711-1787),3  Johann  Christian  Wolf  (1679-1754) 

84,  1881 ;  see  also  H.  DUREGE,  Theorie  der  Funktionen,  Leipzig,  1882, 
p.  226  ;  an  English  translation  of  this  work,  made  by  G.  E.  FISHER  and 
I.  J.  SCHWATT,  has  appeared. 

1  S.  GUNTHER,  Math.  Unt.  im  d.  Mittela.,  pp.  361,  362. 

2  See  LOBATCHEWSKY,   The    Theory  of  Parallels,   transl.    by  G.  B. 
HALSTED,  Austin,  1891,  p.  21,   where  the  theorem   is  proved  that  in 
pseudo-spherical  space,  "The  farther  parallel  lines  are  prolonged  on  the 
side  of  their  parallelism,  the  more  they  approach  one  another." 

3  He  was  professor  at  several  Italian  universities,  was  employed  in 


ELEMENTARY   GEOMETRY  267 

of  Halle,  Thomas  Simpson  (in  the  first  edition  of  his  Elements, 
1747),  John  Bonnycastle  (1750  ?-1821)  of  the  Royal  Military 
Academy  at  Woolwich,  and  others.  It  has  been  used  by  a  few 
American  authors.1  This  definition  is  the  first  of  several  given 
by  E.  Stone  in  his  New  Mathematical  Dictionary,  London,  1743 ; 
in  fact,  "in  the  majority  of  text-books  on  elementary  geome- 
try, from  the  sixteenth  to  the  beginning  of  the  eighteenth 
century,  parallel  lines  are  declared  to  be  lines  that  are 
equidistant  —  which  is,  to  be  sure,  very  convenient." 2  But 
objections  to  this  mode  of  treatment  soon  arose.  As  early  as 
1680  Giordano  da  Bitonto,  in  Italy,  said  that  it  is  inadmissible, 
unless  the  actual  existence  of  equidistant  straight  lines  can 
be  established.  Saccheri  rejects  the  assumption  unceremoni- 
ously.3 

Another  definition,  which  involves  a  tacit  assumption,  de- 
clares parallels  to  be  lines  which  make  the  same  angle  with 
a  third  line.  The  definition  appears  to  have'  originated  in 
France;  it  is  given  by  Pierre  Varignon  (1654-1722)  and 
Etienne  Bezout  (1730-1783),  both  of  Paris.  ,  In  England  it 
was  used  by  Cooley,4  in  the  United  States  by  H.  N.  Robinson. 
A  slight  modification  of  this  is  the  following  definition: 
Parallels  are  lines  perpendicular  to  a  third  line.  This  is  recom- 

various  scientific  duties  by  several  popes,  was  in  London  in  1762,  and 
was  recommended  by  the  Royal  Society  as  a  proper  person  to  be  ap- 
pointed to  observe  the  transit  of  Venus  at  California,  but  the  suppression 
of  the  Jesuit  order,  which  he  had  entered,  prevented  his  acceptance  of 
the  appointment.  See  Penny  Cyclopaedia. 

1  Consult  Teach,  and  Hist,  of  Math,  in  the  U.  8.,  p.  377. 

2  ENGEL  and  STACKEL,  p.  33. 

3  ENGEL  and  STACKEL,  p.  46. 

4  "  MINOS  :  So  far  as  I  can  make  out  Mr.  Cooley  quietly  assumes  that 
a  pair  of  lines  which  make  equal  angles  with  one  line,  do  so  with  all 
lines.     He  might  just  as  well  say  that  a  young  lady,  who  was  inclined  to 
one  young  man,  was  '  equally  and  similarly  inclined  to  all  young  men '  ! 
RHADAMANTHUS :  She  might  'make  equal  angling'  with  them  all  any- 


268  A   HISTORY   OF  MATHEMATICS 

mended  by  the  Italian  G.  A.  Borelli  in  1658,  and  by  the  cele- 
brated French  text-book  writer,  8.  F.  Lacroix.1 

Famous  is  the  definition,  parallel  lines  are  straight  lines  hav- 
ing the  same  direction.  At  first,  this  definition  attracts  us  by 
its  simplicity.  But  the  more  it  is  studied,  the  more  perplexing 
are  the  questions  to  which  it  gives  rise.  Strangely  enough, 
authors  adopting  this  definition  encounter  no  further  diffi- 
culty with  parallel  lines ;  nowhere  do  they  meet  the  necessity 
of  assuming  Euclid's  parallel-postulate  or  any  equivalent  of 
it!  A  question  which  has  perplexed  geometers  for  centuries 
appears  disposed  of  in  a  trice!  In  actual  fact,  there  is, 
perhaps,  no  word  in  mathematics  which,  by  its  apparent  sim- 
plicity, but  real  indefiniteness  and  obscurity,  has  misled  so 
many  able  minds.  In  the  United  States,  as  elsewhere,  this 
definition  has  been  widely  used,  but  for  the  sake  of  sound 
learning,  it  should  be  banished  from  text-books  forever.2 

A  new  definition  of  parallel  lines,  suggested  by  the  principle 
of  continuity,  and  one  of  great  assistance  in  advanced  geome- 
try, though  unsuited  for  elementary  instruction,  is  that  first 

how."  C.  L.  DODGSON,  Euclid  and  His  Modern  Rivals,  London,  1885, 
2d  Ed.,  p.  2  ;  also  p.  62. 

1  S.  F.  LACROIX,  Essais  sur  V enseignement  en  general,  et  sur  celui  des 
mathematiques  en  particulier,  Paris,  1805,  p.  317. 

2  The  objections  to  the  term  "direction"  have  been  ably  set  forth 
so  often,  that  we  need  only  refer  to  some  of  the  articles:  DE  MORGAN'S 
review  of  J.  M.  Wilson's  Geometry,  Athenceum,  July  18,  1868 ;  E.  L. 
RICHARDS  in  Educational  Review,  Vol.  III.,  1892,  p.  32  ;  Seventh  General 
Report  (1891)  of  the  A.  I.  G.  T.,  pp.  36-42 ;  G.  B.  HALSTED,  The  Science 
Absolute  of  Space  by  JOHN  BOLYAI,  4th  Ed.,  1896,  pp.   63-71;  G.  B. 
HALSTED  in  Educational  Review,  Sept.,  1893,  p.  153 ;  C.  L.  DODGSON, 
Euclid  and  His  Modern  Rivals,  London ;  H.  MULLER,  Besitzt  die  heutige 
Schulgeometrie  noch  die   Vorzuge  des  Euklidischen    Originals  f  Metz, 
1889,  p.  7  ;  Zeitschrift  fUr  mathematischen  und  naturwissenschaftlichen 
Unterricht,  Teubner  in  Leipzig,  XVI.,  p.  407  ;   Teach,  and  Hist,  of  Math, 
in  the    U.   S.,  pp.  381-383  ;   Monist,  1892   (Review  of  E.   T.   DIXON'S 
"Foundations  of  Geometry"). 


ELEMENTARY   GEOMETRY  269 

given  by  John  Kepler  (1604)  and  Girard  Desargues  (1639): 
Lines  are  parallel  if  they  have  the  same  infinitely  distant 
point  in  common.  A  similar  idea  is  expressed  by  E.  Stone  in 
his  dictionary  thus :  "  If  A  be  a  point  without  a  given  indefi- 
nite right  line  CD ;  the  shortest  line,  as  AB,  that  can  be  drawn 
from  A  to  it,  is  perpendicular ;  and  the  longest,  as  EA,  is 
parallel  to  CD." 

Quite  a  number  of  substitutes  for  Euclid's  parallel-postulate 
differing  in  form,  but  not  in  essence,  have  been  suggested 
at  various  times.  In  the  evening  of  July  11,  1663,  John 
Wallis  delivered  a  lecture  at  Oxford  on  the  parallel-postulate.1 
He  recommends  in  place  of  Euclid's  postulate :  To  any  triangle 
another  triangle,  as  large  as  you  please,  can  be  drawn,  which  is 
similar  to  the  given  triangle.  Sacclieri  showed  that  Euclidean 
geometry  can  be  rigidly  developed  if  the  existence  of  one  tri- 
angle, unequal  but  similar  to  another,  be  presupposed.  Lam- 
bert makes  similar  remarks.  Wallis's  postulate  was  again 
proposed  for  adoption  by  L.  Carnot  and  Laplace,  and  more 
recently  by  J.  Delboeuf.2  Alexis  Claude  Clairaut  (1713-1765), 
a  famous  French  mathematician,  wrote  an  elementary  geom- 
etry, in  which  he  assumes  the  existence  of  a  rectangle,  and  from 
this  substitute  for  Euclid's  postulate  develops  the  elementary 
theorems  with  great  clearness.  Other  equivalent  postulates 
are  the  following :  a  circle  can  be  passed  through  any  three 
points  not  in  the  same  straight  line  (due  to  W.  Bolyai) ;  the 
existence  of  a  finite  triangle  whose  angle-sum  is  two  right 
angles  (due  to  Legendre) ;  through  every  point  within  an  angle 
a  line  can  be  drawn  intersecting  both  sides  (due  to  J.  F. 
Lorenz  (1791),  and  Legendre) ;  in  any  circle,  the  inscribed  equi- 
lateral quadrangle  is  greater  than  any  one  of  the  segments 

1  See  Opera,  Vol.  II.,  665-678.     In  German  translation  it  is  given  by 
ENGEL  and  STACKEL,  pp.  21-30. 

2  ENGEL  and  STACKEL,  p.  19. 


270  A   HISTORY   OF   MATHEMATICS 

which  lie  outside  of  it  (due  to  C.  L.  Dodgson) ;  two  straight 
lines  which  intersect  each  other  cannot  be  both  parallel 
to  the  same  straight  line  (used  by  John  Playfair).1  Of  all 
these  substitutes,  only  the  last  has  met  with  general  favour. 
Playfair  adopted  it  in  his  edition  of  Euclid,  and  it  has 
been  generally  recognized  as  simpler  than  Euclid's  parallel- 
postulate. 

Until  the  close  of  the  first  quarter  of  the  nineteenth  century 
it  was  widely  believed  by  mathematicians  that  Euclid's  parallel- 
postulate  could  be  proved  from  the  other  assumptions  and  the 
definitions  of  geometry.  We  have  already  referred  to  such 
efforts  made  by  Ptolemy  and  Nasir  Eddin.  We  refrain  from 
discussing  proofs  in  detail.  They  all  fail,  either  because  an 
equivalent  assumption  is  implicitly  or  explicitly  made,  or 
because  the  reasoning  is  otherwise  fallacious.  On  this  slippery 
ground  good  and  bad  mathematicians  alike  have  fallen.  We 
are  told  that  the  great  Lagrange,  noticing  that  the  formulae  of 
spherical  trigonometry  are  not  dependent  upon  the  parallel- 
postulate,  hoped  to  frame  a  proof  on  this  fact.  Toward  the 
close  of  his  life  he  wrote  a  paper  on  parallel  lines  and  began 
to  read  it  before  the  Academy,  but  suddenly  stopped  and 
said :  "  II  faut  que  j'y  songe  encore  " 2  (I  must  think  it  over 
again) ;  he  put  the  paper  in  his  pocket  and  never  afterwards 
publicly  recurred  to  it. 

The  researches  of  Adrien  Marie  Legendre  (1752-1833)  are 
interesting.  Perceiving  that  Euclid's  postulate  is  equivalent 
to  the  theorem  that  the  angle-sum  of  a  triangle  is  two  right 
angles,  he  gave  this  an  analytical  proof,  which,  however, 
assumes  the  existence  of  similar  figures.  Legendre  was  not 
satisfied  with  this.  Later,  he  proved  satisfactorily  by  assum- 
ing lines  to  be  of  indefinite  extent,  that  the  angle-sum  can- 

1  Consult  G.  B.  HALSTED'S  Bolyai,  4th  Ed.,  pp.  64,  66. 

2  ENGEL  and  STACKEL,  p.  212. 


ELEMENTARY   GEOMETRY  271 

not  exceed  two  right  angles,  but  could  not  prove  that  it  cannot 
fall  short  of  two  right  angles.  In  1823,  in  the  twelfth  edition 
of  his  Elements  of  Geometry,  he  thought  he  had  a  proof  for 
the  second  part.  Afterwards,  however,  he  perceived  its  weak- 
ness, for  it  rested  on  the  new  assumption  that  through  any 
point  within  an  angle  a  line  can  be  drawn,  cutting  both  sides 
of  the  angle.  In  1833  he  published  his  last  paper  on  parallels, 
in  which  he  correctly  proves  that,  if  there  be  any  triangle 
the  sum  of  whose  angles  is  two  right  angles,  then  the  same 
must  be  true  of  all  triangles.  But  in  the  next  step,  to  show 
rigorously  the  actual  existence  of  such  a  triangle,  his  demon- 
stration failed,  though  he  himself  thought  he  had  finally  settled 
the  whole  question.1  As  a  matter  of  fact  he  had  not  gotten 
quite  so  far  as  had  Saccheri  one  hundred  years  earlier.2 
Moreover,  before  the  publication  of  his  last  paper,  a  Russian 
mathematician  had  taken  a  step  which  far  transcends  in 
boldness  and  importance  anything  Legendre  had  done  on  this 
subject. 

As  with  the  problem  of  the  quadrature  of  the  circle,  so 
with  the  parallel-postulate :  after  numberless  failures  on  the 
part  of  some  of  the  best  minds  to  resolve  the  difficulty, 
certain  shrewd  thinkers  began  to  suspect  that  the  postulate 

1  A  novel  attempt  to  prove  the  angle-sum  of  a  triangle  to  be  two  right 
angles  was  made  in  1813  by  John  Playfair,  in  his  Elements  of  Geometry. 
See  the  edition  of  1855,  Philadelphia,  pp.  295,  296.     It  consists,  briefly, 
in  laying  a  straight  edge  along  one  of  the  sides  of  the  figure,  and  then 
turning  the  edge  round  so  as  to  coincide  with  each  in  turn.     Then  the 
edge  is  said  to  have  described  angles  whose  sum  is  four  right  angles. 
The  same  argument  proves  the  angle-sum  of  spherical  triangles  to  be  two 
right  angles,  a  result  which  we  know  to  be  wrong.     It  is  to  be  regretted 
that  in  two  American  text-books,  just  published,  this  argument  is  repro- 
duced, thus  giving  this  famous  heresy  new  life.     For  the  exposition  of 
the  fallacy  see  G.  B.  HALSTED'S  4th  Ed.  of  BOLYAI'S  Science  Absolute  oj 
Space,  pp.  63-71  ;  Nature,  Vol.  XXIX.,  1884,  p.  453. 

2  ENGEL  and  STACKEL,  pp.  212,  213. 


272  A   HISTORY  OF   MATHEMATICS 

did  not  admit  of  proof.  This  scepticism  appears  in  various 
writings.1 

If  it  required  courage  on  the  part  of  Euclid  to  place  his 
parallel-postulate,  so  decidedly  unaxiomatic,  among  his  other 
postulates  and  common  notions,  even  greater  courage  was  de- 
manded to  discard  a  postulate  which  for  two  thousand  years 
had  been  the  main  corner  stone  of  the  geometric  structure. 
Yet  several  thinkers  of  the  eighteenth  and  nineteenth  cen- 
turies displayed  that  independence  of  thought,  so  essential  to 
great  discoveries. 

While  Legendre  still  endeavoured  to  establish  the  parallel- 
postulate  by  rigorous  proof,  Nicholaus  Ivanovttch  Lobatchewsky 
(1793-1856)  brought  out  a  publication  which  assumed  the 
contradictory  of  that  axiom.  Lobatchewsky's  views  on  the 
foundation  of  geometry  were  first  made  public  in  a  discourse 
before  the  physical  and  mathematical  faculty  of  the  University 
of  Kasan  (of  which  he  was  then  rector),  and  first  printed  in 
the  Kasan  Messenger  for  1829,  and  then  in  the  Gelehrte 
Scliriften  der  Universitdt  Kasan,  1836-1838,  under  the  title 
"New  Elements  of  Geometry,  with  a  complete  theory  of 
parallels."  This  work,  in  Eussian,  not  only  remained  un- 
known to  foreigners,  but  attracted  no  notice  at  home.  In  1840 
he  published  in  Berlin  a  brief  statement  of  his  researches. 
The  distinguishing  feature  of  this  "Imaginary  Geometry" 
of  Lobatchewsky  is  that  through  a  point  an  indefinite  number 
of  lines  can  be  drawn  in  a  plane,  none  of  which  cut  a  given 
line  in  the  plane,  and  that  the  sum  of  the  angles  in  a  triangle 
is  variable,  though  always  less  than  two  right  angles.2 

1  ENGEL  and  STACKEL,  pp.  140,  141,  213-215. 

2  LOBATCHEWSKY'S    Theory    of  Parallels    has   been    translated    into 
English  by  G.  B.  HALSTED,  Austin,  1891.     It  covers  only  forty  pages. 
G.  B.  Halsted  has  also  translated  Professor  A.  VASILIEV'S  Address  on 
the  life  and  researches  of  Lobatchewsky,  Austin,  1894. 


ELEMENTARY   GEOMETRY  273 

A  similar  system  of  geometry  was  devised  by  the  Bolyais  in 
Hungary  and  called  by  them  "  absolute  geometry."  Wolfgang 
Bolyai  de  Bolya  (1775-1856)  studied  at  Jena,  then  at  Gottingen, 
where  he  became  intimate  with  young  Gauss.  Later,  he  be- 
came professor  at  the  Reformed  College  of  Maros-Vasarhely, 
where  for  forty-seven  years  he  had  for  his  pupils  most  of  the 
present  professors  in  Transylvania.  Clad  in  old  time  planter's 
garb,  he  was  truly  original  in  his  private  life  as  well  as  in  his 
mode  of  thinking.  He  was  extremely  modest.  No  monument, 
said  he,  should  stand  over  his  grave,  only  an  apple-tree,  in 
memory  of  the  three  apples :  the  two  of  Eve  and  Paris,  which 
made  hell  out  of  earth,  and  that  of  Newton,  which  elevated 
the  earth  again  into  the  circle  of  heavenly  bodies.1  His  son, 
Johann  Bolyai  (1802-1860)  was  educated  for  the  army,  and 
distinguished  himself  as  a  profound  mathematician,  an  im- 
passioned violin-player,  and  an  expert  fencer.  He  once 
accepted  the  challenge  of  thirteen  officers  on  condition  that 
after  each  duel  he  might  play  a  piece  on  his  violin,  and  he 
vanquished  them  all.2 

Wolfgang  Bolyai's  chief  mathematical  work,  the  Tentamen, 
appeared  in  two  volumes,  1832-1833.  The  first  volume  is 
followed  by  an  appendix  written  by  his  son  Johann  on  The 
Science  Absolute  of  Space.  Its  twenty-six  pages  make  the 
name  of  Johann  Bolyai  immortal.  Yet  for  thirty-six  years 
this  appendix,  as  also  Lobatchewsky's  researches,  remained  in 
almost  entire  oblivion.  Finally  Richard  Baltzer,  of  Giessen, 
in  1867,  called  the  attention  of  mathematicians  to  these  won- 

1  F.    SCHMIDT,    "  Aus  dem  Leben   zweir  ungarischer  Mathematiker 
Johann  und  Wolfgang  Bolyai  von  Bolya,"  GRUNERT'S  Archiv  der  Mathe- 
matik  und  Physik,  48  :  2,  1868. 

2  For  additional  biographical  detail,  see  G.  B.  HALSTED'S  translation 
of  JOHN  BOLYAI'S  The  Science  Absolute  of  Space,  4th  Ed.  1896.     Dr. 
Halsted  is  about  to  issue  The  Life  of  Bolyai,  containing  the  Autobiog- 
raphy of  Bolyai  Farkas,  and  other  interesting  information. 


274  A   HISTORY   OF   MATHEMATICS 

derful  studies.  In  1894  a  monumental  stone  was  placed  on 
the  long-neglected  grave  of  Johann  Bolyai  in  Maros-Vasarhely. 
In  the  years  1893-1895  a  Lobatchewsky  fund  was  secured 
through  contributions  of  scientific  men  in  all  countries,  which 
goes  partly  toward  founding  an  international  prize  for  re- 
search in  geometry,  and  partly  towards  erecting  a  bust  of 
Lobatchewsky  in  the  park  in  front  of  the  university  building 
in  Kasan. 

But  the  Russian  and  Hungarian  mathematicians  were  not 
the  only  ones  to  whom  the  new  geometry  suggested  itself. 
When  Gauss  saw  the  Tentamen  of  the  elder  Bolyai,  his  former 
room-mate  at  Gottingen,  he  was  surprised  to  find  worked  out 
there  what  he  himself  had  begun  long  before,  only  to  leave  it 
after  him  in  his  papers.  His  letters  show  that  in  1799  he  was 
still  trying  to  prove  a  priori  the  reality  of  Euclid's  system,  but 
later  he  became  convinced  that  this  was  impossible.  Many 
writers,  especially  the  Germans,  assume  that  both  Lobatchew- 
sky and  Bolyai  were  influenced  and  encouraged  by  Gauss,  but 
no  proof  of  this  opinion  has  yet  been  presented.1 

Recent  historical  investigation  has  shown  that  the  theories 
of  Lobatchewsky  and  Bolyai  were,  in  part,  anticipated  by  two 
writers  of  the  eighteenth  century,  Oeronimo  Saccheri  (1667- 
1733),  a  Jesuit  father  of  Milan,  and  Johann  Heinrich  Lambert 
(1728-1777),  a  native  of  Muhlhausen,  Alsace.  Both  made 
researches  containing  definitions  of  the  three  kinds  of  space, 
now  called  the  non-Euclidean,  Spherical,  and  the  Euclidean 
geometries.2 

1  ENGEL  and  STACKEL,  pp.  242,  243 ;  G.  B.  HALSTED,  in  Science,  Sept. 
6,  1895. 

2  The  researches  of  Wallis,  Saccheri,  and  Lambert,  together  with  a  full 
history  of  the  parallel  theory  down  to  Gauss,  are  given  by  ENGEL  and 
STACKEL.     See  also  G.  B.  HALSTED'S  "The  non-Euclidean  Geometry  in- 
evitable" in  the  Monist,  July,  1894.     Space  does  not  permit  us  to  speak 
of  the  later  history  of  non-Euclidean  geometry.     We  commend  to  our 


ELEMENTARY   GEOMETRY  275 

We  have  touched  upon  the  subject  of  non-Euclidean  geome- 
try, because  of  the  great  light  which  these  studies  have 
thrown  upon  the  foundations  of  geometric  theory.  Thanks 
to  these  researches,  no  intelligent  author  of  elementary  text- 
books will  now  attempt  to  do  what  used  to  be  attempted: 
prove  the  parallel-postulate.  We  know  at  last  that  such  an 
attempt  is  futile.  Moreover,  many  trains  of  reasoning  are  now 
easily  recognized  as  fallacious  by  one  who  sees  with  the  eyes 
of  Lobatchewsky  and  Bolyai.  Possessing,  as  we  do,  English 
translations  of  their  epoch-making  works,  no  progressive 
teacher  of  geometry  in  High  School  or  College  —  certainly  no 
author  of  a  text-book  on  geometry  —  can  afford  to  be  ignorant 
of  these  results. 

IV.  Text-books  on  Elementary  Geometry.  —  The  history  of 
the  evolution  of  geometric  text-books  proceeds  along  different 
lines  in  the  various  European  countries.  About  the  time  when 
Euclid  was  translated  from  the  Arabic  into  Latin,  such  intense 
veneration  came  to  be  felt  for  the  book  that  it  was  considered 
sacrilegious  to  modify  anything  therein.  Still  more  pronounced 
was  this  feeling  toward  Aristotle.  Thus  we  read  of  Petrus 
Ramus  (1515-1572)  in  France  being  forbidden  on  pain  of 
corporal  punishment  to  teach  or  write  against  Aristotle.  This 
royal  mandate  induced  Ramus  to  devote  himself  to  mathe- 

readers  Halsted's  translations  of  Lobatchewsky  and  Bolyai.  Georg 
Friedrich  Bernhard  Riemann's  wonderful  dissertation  of  1854,  entitled 
Ueber  die  Hypothesen  welche  der  G-eometrie  zu  Grunde  liegen  is  trans- 
lated by  W.  K.  Clifford  in  Nature,  Vol.  8,  pp.  14-17,  36-37.  Riemann 
developed  the  notion  of  w-ply  extended  magnitude.  Helmholtz's  lecture, 
entitled  "Origin  and  Significance  of  Geometrical  Axioms"  is  contained 
in  a  volume  of  Popular  Lectures  on  Scientific  Subjects,  translated  by 
E.  ATKINSON,  New  York,  1881.  Consult  also  the  works  of  W.  K.  Clifford. 
The  bibliography  of  non-Euclidean  geometry  and  hyperspace  is  given  by 
G.  B.  HALSTED  in  the  American  Journal  of  Mathematics,  Vols.  I.  and  II. 
Readers  may  be  interested  in  Flatland,  a  Bomance  of  Many  Dimensions, 
published  by  Roberts  Brothers,  Boston,  1885. 


276  A   HISTORY   OF   MATHEMATICS 

matics ;  lie  brought  out  an  edition  of  Euclid,  and  here  again 
displayed  his  bold  independence.  He  did  not  favour  inves- 
tigations on  the  foundations  of  geometry ;  he  believed  that  it 
was  not  at  all  desirable  to  carry  everything  back  to  a  few 
axioms;  whatever  is  evident  in  itself,  needs  no  proof.  His 
opinion  on  mathematical  questions  carried  great  weight.  His 
views  respecting  the  basis  of  geometry  controlled  French 
text-books  down  to  the  nineteenth  century.  In  no  other  civil- 
ized country  has  Euclid  been  so  little  respected  as  in  France. 
A  conspicuous  example  of  French  opinion  on  this  matter  is 
the  text  prepared  by  Alexis  Claude  Clairaut  in  1741.  He  con- 
demns the  profusion  of  self-evident  propositions,  saying  in  his 
preface,  "  It  is  not  surprising  that  Euclid  should  give  himself 
the  trouble  to  demonstrate  that  two  circles  which  intersect 
have  not  the  same  centre;  that  a  triangle  situated  within 
another  has  the  sum  of  its  sides  smaller  than  that  of  the  sides 
of  the  triangle  which  contains  it.  That  geometer  had  to  con- 
vince' obstinate  sophists,  who  gloried  in  denying  the  most 
evident  truths  .  .  . ;  but  in  our  day  things  have  changed  face ; 
all  reasoning  about  what  mere  good  sense  decides  in  advance 
is  now  a  pure  waste  of  time  and  fitted  only  to  obscure  the 
truth  and  to  disgust  the  reader."  This  book,  precisely  antip- 
odal to  Euclid,  has  contributed  much  toward  moulding  opin- 
ions on  geometric  teaching,  but  otherwise  it  did  not  enjoy  a 
great  success.1  Similar  views  were  entertained  by  Etienne 
Bezout  (1730-1783),  whose  geometric  works,  like  Clairaut's; 
are  deficient  in  rigour.  Somewhat  more  methodical  was 
Sylvestre  Francois  Lacroix  (1765-1843).  But  the  French 
geometry  which  enjoyed  the  most  pronounced  success,  both 
at  home  and  in  other  countries,  is  that  of  Adrien  Marie 
Legendre,  first  brought  out  in  1794.  It  is  interesting  to  note 

1  See  article  "  Ge'ome'trie  "  in  the  Grand  Dictionnaire  Universal  du 
XIX*  Siecle  par  PIERRE  LAROUSSE. 


ELEMENTARY   GEOMETRY  277 

Loria's  estimate  of  Legendre's  Elements  de  g&om&rie.1  "  They 
deserved  it  [success],  since,  as  regards  form,  if  they  can  com- 
pete with  Euclid  in  clearness  and  precision  of  style,  they  are 
superior  to  him  in  the  complex  harmony  which  gives  them  the 
appearance  of  a  beautiful  edifice,  divided  into  two  symmetrical 
parts,  assigned,  the  one  to  the  geometry  of  the  plane  and  the 
other  to  the  geometry  of  space ;  and  since  in  regard  to  matter, 
they  surpass  Euclid  by  being  richer  in  material  and  better  in 
certain  particulars.  But  the  great  French  analyst,  in  writing 
of  geometry,  could  not  forget  his  own  favoured  studies,  so  that 
in  his  hands  geometry  became  a  vassal  of  arithmetic,  from 
which  he  borrowed  a  few  ratiocinations  and  even  some  of  his 
nomenclature,  and  took  from  its  dominions  the  whole  theory 
of  proportions.  If  it  is  added  that,  while  Euclid  avoids  the 
use  of  any  figure,  the  construction  of  which  is  unknown  to 
the  reader,  Legendre  uses  without  scruple  the  so-called  'hypo- 
thetical constructions/  and  that  he  gave  the  preference  to  that 
unfortunate  definition  of  a  straight  line  [used,  moreover,  even 
by  Kant]  as  a  minimum  line ;  there  is  sufficient  argument  to 
support  the  fact  that  Legendre's  edifice  was  not  long  in  show- 
ing itself  of  a  solidity  incomparably  inferior  to  its  beauty." 2 

Thus  we  see  that  French  writers,  influenced  by  their  views 
respecting  methods  of  teaching,  by  their  belief  that  what  is 
apparent  to  the  eye  may  be  accepted  by  the  young  student 
without  proof,  and  by  their  general  desire  to  make  geometry 
easier  and  more  palatable,  allowed  themselves  to  depart  a  long 
way  from  the  practice  of  Euclid.  But  Legendre  is  more  strict 
than  Clairaut;  the  reaction  against  Clairaut's  views  became 
more  and  more  pronounced,  and  about  the  middle  of  the  nine- 

1  LORIA,  Delia  variafortuna  di  Euclide,  Roma,  1893,  p.  10. 

2  An  American  edition  of  Brewster's  translation  of  Legendre's  geom- 
etry was  brought  out  in  1828  by  Charles  Davies  of  West  Point.    John 
Farrar  of  Harvard  College  issued  a  new  translation  in  1830. 


278  A   HISTORY   OF  MATHEMATICS 

teenth  century,  Jean  Marie  Constant  Duhamel  (1797-1872)  and 
J.  Houel1  began  to  institute  comparisons  between  Legendrian 
and  Euclidean  methods  in  favour  of  the  latter,  recommending 
the  return  to  a  modified  Euclid.  Duhamel  proclaimed  the 
method  of  limits  to  be  the  only  rigorous  one  by  which  the  use  of 
the  infinite  can  be  introduced  into  geometry,2  and  contributed 
much  toward  imparting  to  that  method  a  clear  and  unobjection- 
able form.  Houel,  professor  at  Bordeaux,  while  expressing 
his  regret  that  Euclid's  Elements  had  fallen  into  disuse  in 
France,  did  not  recommend  the  adoption  of  Euclid  unmodified. 

While  Duhamel's  and  Houel's  desired  return  to  Euclidean 
methods  did  not  take  place,  the  discussion,  nevertheless,  led 
to  improvements  in  geometric  texts.  The  works  which  in 
France  now  enjoy  the  greatest  favour  are  those  of  E.  Rouche 
and  C.  de  Comberousse,  and  of  Ch.  Vacquant*  The  treatment 
of  incommensurables  is  by  the  method  of  limits,  against  which 
the  cry  "lack  of  rigour"  cannot  be  raised.  In  appendices 
considerable  attention  is  given  to  projective  geometry,  but 
Loria  remarks  that  the  old  and  the  new  in  geometry  are  here 
not  united  into  a  coherent  whole,  but  merely  mixed  in  dis- 
jointed fashion.4 

Italy,  the  land  where  once  Euclid  was  held  in  high  venera- 
tion, where  Saccheri  wrote,  in  1733,  Eudides  ab  omni  naevo 
vindicatus  (Euclid  vindicated  from  every  flaw),  finally  discarded 
her  Euclid  and  departed  from  the  spirit  of  his  Elements.  In 
1867  Professors  Cremona  of  Milan  and  Battaglini  of  Naples 

1  DUHAMEL,  Des  methodes  dans  les  sciences  de  raisonnement  (five  vol- 
umes), 1866-1872,  II.,  326;  HOUEL,  Essai  sur  les  principes  fondament 
aux  de  la  geometric  elementaire  ou  commentaire  sur  les  XXXII.  premi- 
eres propositions  des  elements  d^Euclide  (Paris,  1867) . 

2  LORIA,  op.  cit.,  p.  11.  8  LORIA,  op.  cit.,  p.  12. 

4  An  elementary  geometry  of  great  merit,  modelled  after  French 
works,  particularly  that  of  Roucne"  and  Comberousse,  was  prepared  in 
1870  by  William  Chauvenet  of  Washington  University  in  St.  Louis. 


ELEMENTARY   GEOMETRY  279 

were  members  of  a  special  government  commission  to  inquire 
into  the  state  of  geometrical  teaching  in  Italy.  They  found 
it  to  be  so  unsatisfactory  and  the  number  of  bad  text-books  so 
great  and  so  much  on  the  increase,  that  they  recommended  for 
classical  schools  the  adoption  of  Euclid  pure  and  simple,  even 
though  Cremona  admitted  that  Euclid  is  faulty.1  This  recom- 
mendation became  law.  Later  the  use  of  Euclid's  text  was 
replaced  by  that  of  other  works,  prepared  on  analogous  plans.2 
Thereupon  appeared  after  the  amended  Euclidean  model,  as 
distinguished  from  the  Legendrian,  the  meritorious  work  by 
A.  Sannia  and  E.  D'  Ovidio.  Later  came  the  Elementi  di 
geometria  of  Biccardo  de  Paolis  and  the  Fondamenti  di  geom- 
etria  of  Guiseppe  Veronese. 

The  'high  esteem  in  which  Euclid  was  held  in  Germany 
during  early  times  is  demonstrated  by  many  facts.  About  the 
close  of  the  eighteenth  century  Abraham  Gotthelf  Kastner 
remarked  that  "the  more  recent  works  on  geometry  depart 
from  Euclid,  the  more  they  lose  in  clearness  and  thorough- 
ness." This  points  to  a  falling  off  from  the  Euclidean  model. 
The  most  popular  texts  in  use  about  the  middle  of  the  nine- 
teenth century,  and  even  those  of  the  present  time,  are,  from 
a  scientific  point  of  view,  anything  but  satisfactory.  Thus 
in  H.  B.  Liibsen's  Elementar-Geometrie,  14th  Ed.,  Leipzig, 
1870,  written  in  the  country  which  produced  Gauss,  and  at 
a  time  when  Lobatchewsky's  and  Bolyai's  immortal  works 
had  been  published  41  and  37  years  respectively,  we  still  find 
(p.  52)  a  proof  of  the  parallel-postulate  ! 3  If  we  bear  in  mind 
that  geometry  deals  with  continuous  magnitudes,  in  which  com- 

1  HIRST,  address  in  First  Annual  Report,  A.  I.  G.  T.,  1871. 

2  LORIA,  op.  cit.,  p.  15. 

8  Another  illustration  of  the  confusion  which  prevailed  regarding  the 
foundations  of  geometry,  long  after  Lobatchewsky,  is  found  in  Thomas 
Peronnet  Thompson's  Geometry  without  Axioms,  3d  Ed.,  1830,  in  which 
he  endeavoured  to  "  get  rid  "  of  axioms  and  postulates  1 


280  A   HISTORY   OF   MATHEMATICS 

mensurability  is  exceptional,  it  is  somewhat  startling  that  Liib- 
sen  should  make  no  mention  of  incommensurables.  Another 
work  which  has  been  used  for  several  decennia  is  Karl  Koppe's 
Planimetrie,  4th  Ed.,  Essen,  1852.  It  speaks  of  parallels  as 
having  the  same  direction,  and  fails  to  consider  incommen- 
surability excepting  once  in  a  foot-note.  Kambly's  inferior 
work  appeared  in  1884  in  the  74th  edition.1  The  subject  of 
geometric  teaching  has  been  much  discussed  in  Germany. 
Some  excellent  text-books  have  been  written,  but  they  do  not 
appear  to  be  popular.  Among  the  better  works  are  those 
of  Baltzer,  Schlegel,  H.  Mtiller,  Kruse,  Worpitzky,  Henrici 
and  Treutlein.2  Most  of  these  seem  to  be  dominated  by 
Euclidean  methods.3  One  of  the  questions  debated  in  Ger- 
many and  elsewhere  is  that  pertaining  to  the  rigidity  of  fig- 
ures. While  Euclid  sometimes  moves  a  figure  as  a  whole,  he 
never  permits  its  parts  to  move  relatively  to  one  another. 
With  him,  figures  are  "rigid."  In  many  modern  works  this 
practice  is  abandoned.  For  instance,  we  often  allow  an  angle 
to  be  generated  by  rotation  of  a  line,4  whereby  we  arrive  at 
the  notion  of  a  "  straight  angle  "  (not  used  by  Euclid),  which 
now  competes  with  the  right  angle  as  a  unit  for  angular 
measure.  The  abandonment  of  rigidity  brings  us  in  closer 
touch  with  the  modern  geometry,  and  is,  we  think,  to  be 
recommended,  provided  that  we  proceed  with  circumspection. 
Pure  projective  geometry  needs  neither  motion  nor  rigidity. 

1  A.  ZIWET  in  Bulletin  N.  Y.  Math.  Soc.,  1891,  p.  6.  Ziwet  here  re- 
views a  book  containing  much  information  on  the  movement  in  Germany, 
viz.  H.  SCHOTTEN,  Inhalt  und  Methods  des  planimetrischen  Unterrichts. 
The  reader  will  profit  by  consulting  HOFFMANN'S  Zeitschrift  fur  mathe- 
matischen  und  naturwissenschaftlichen  Unterricht,  published  by  Teub- 
ner,  Leipzig.  2  LORIA,  p.  19. 

8  See  LORIA,  p.  19,  SCHOTTEN,  op.  cit.  ;  H.  MULLER,  Besitzt  die  heutige 
Schulgeometrie  nochdie  Vorzugedes  Euklidischen  Originals  f  Metz,  1889. 

4  H,  MULLER,  op.  c#.,  p.  2. 


ELEMENTARY   GEOMETRY  281 

Preparatory  courses  in  intuitive  geometry,  in  connection  with 
geometric  drawing,  have  been  widely  recommended  and  adopted 
in  Germany. 

England  has  been  the  home  of  conservatism  in  geometric 
teaching.  Wherever  in  England  geometry  has  been  taught, 
Euclid  has  been  held  in  high  esteem.  It  appears  that  the 
first  to  rescue  the  Elements  from  the  Moors  and  to  bring  it  out 
in  Latin  translation  was  an  Englishman.  The  first  English 
translation  (Billingsley's)  was  printed  in  1570.  Before  this, 
Robert  Eecorde  made  a  translation,  but  it  was  probably  never 
published.1  Elsewhere  we  have  spoken  of  mediaeval  geomet- 
rical teaching  at  Oxford.  It  was  carried  on  with  very  limited 
success.  About  1620  Sir  Henry  Savile  (1549-1622),  warden 
of  Merton  College,  endeavoured  to  create  an  interest  in  mathe- 
matical studies  by  giving  a  course  of  lectures  on  Greek 
geometry.  These  were  published  in  1621.  On  concluding 
the  course  he  used  the  following  language :  "  By  the  grace  of 
God,  gentlemen  hearers,  I  have  performed  my  promise;  I 
have  redeemed  my  pledge.  I  have  explained,  according  to  my 
ability,  the  definitions,  postulates,  axioms,  and  the  first  eight 
propositions  of  the  Elements  of  Euclid.  Here,  sinking  under 
the  weight  of  years,  I  lay  down  my  art  and  my  instruments."  2 
It  must  be  remembered  that  at  this  time  scholastic  learning 
and  polemical  divinity  held  sway.  Savile  ranks  among  those 
who  laboured  for  the  revival  of  true  knowledge.  He  founded 
two  professorships  at  Oxford,  one  of  geometry  and  the  other 
of  astronomy,  and  endowed  each  with  a  salary  of  £150  per 
annum.  In  the  preamble  to  the  deed  by  which  he  annexed 
this  salary  he  says  that  geometry  was -almost  totally  abandoned 
and  unknown  in  England. 

1  W.  W.  R.  BALL,  Maths,  at  Cambridge,  p.  18. 

2  Translated  from  the  Latin  by  W.   WHEWELL,  in  his  Hist,  of  the 
Induct.  Sciences,  Vol.  I.,  New  York,  1858,  p.  205. 


282  A  HISTORY  OF  MATHEMATICS 

Savile  delivered  thirteen  lectures  in  his  course,  in  which 
philological  and  historical  questions  received  more  attention 
than  did  geometry  itself.1  In  one  place  he  says,  "In  the 
beautiful  structure  of  geometry  there  are  two  blemishes,  two 
defects ;  I  know  no  more."  These  "  blemishes  "  are  the  theory 
of  parallels  and  the  theory  of  proportion.  At  that  time 
Euclid's  parallel-postulate  was  objected  to  as  unaxiomatic 
and  requiring  demonstration.  We  now  know  by  the  light  de- 
rived from  non-Euclidean  geometry  that  it  is  a  pure  as- 
sumption, incapable  of  proof;  that  a  geometry  exists  which 
assumes  its  contradictory;  that  it  separates  the  Euclidean 
from  the  pseudo-spherical  geometry.  No  "  blemish,"  there- 
fore, exists  on  this  point  in  Euclid's  Elements.  The  second 
"  blemish  "  referred  to  the  sixth  statement  in  Book  V.2  Peda- 
gogically,  this  fifth  book  is  still  criticised,  on  account  of  its 
excessive  abstruseness  for  young  minds;  but  scientifically 
(aside  from  certain  unimportant  emendations  which  Robert 
Simson  thought  necessary)  this  book  is  now  regarded  as  one 
of  exceptional  merit. 

During  the  seventeenth  and  eighteenth  centuries  a  consider- 
able number  of  English  editions  of  Euclid  were  brought  out. 
These  were  supplanted  after  1756  by  Robert  Simson's  Euclid. 
In  the  early  part  of  the  nineteenth  century  Euclid  was  still 
without  a  rival  in  Great  Britain,  but  the  desirability  of  modi- 
fication arose  in  some  minds.  As  early  as  1795,  John  Play- 
fair  (1748-1819)  brought  out  at  Edinburgh  his  Elements 
of  Geometry,  containing  the  first  six  books  of  Euclid  and  a 
supplement,  embracing  an  approximate  quadrature  of  the 
circle  (i.e.  the  computation  of  TT)  and  a  book  on  solid  geom- 
etry drawn  from  other  sources.  In  the  fifth  book  of  Euclid, 
Playfair  endeavours  to  remove  Euclid's  diffuseness  of  style  by 

1  CANTOR,  II.,  664.  2  ENGEL  and  STACKEL,  p.  47. 


ELEMENTARY   GEOMETRY  283 

introducing  the  language  of  algebra.  Playfair  also  introduces 
a  new  parallel-postulate,  simpler  than  Euclid's.  Gradually 
the  advantage  of  more  pronounced  deviations  from  Euclid's 
text  was  expressed.  In  1849  De  Morgan  pointed  out  defects 
in  Euclid.1  About  twenty  years  later  Wilson2  and  Jones3 
expressed  the  desirability  of  abandoning  the  Euclidean  method. 
Finally  in  1869  one  of  the  two  greatest  mathematicians  in 
England  raised  his  powerful  voice  against.  Euclid.  J.  J.  Syl- 
vester said,  "I  should  rejoice  to  see  ...  Euclid  honourably 
shelved  or  buried  ' deeper  than  e'er  plummet  sounded'  out 
of  the  schoolboy's  reach." 4  These  attacks  on  Euclid  as  a  school- 
book  brought  about  no  immediate  change  in  geometrical  teach- 
ing. The  occasional  use  of  Brewster's  translation  of  Legendre 
or  of  Wilson's  geometry  no  more  indicated  a  departure  from 
Euclid  than  did  the  occasional  use,  in  the  eighteenth  century, 

1  LORIA,  op.  cit.,  p.  24,  refers  to  DE  MORGAN  in  the  Companion  to  the 
British  Almanac,  which  we  have  not  seen. 

2  Educational  Times,  1868. 

3  On  the  Unsuitableness  of  Euclid  as  a  Text-book  of  Geometry,  London. 

4  SYLVESTER'S  Presidential  Address  to  the  Math,  and  Phys.  Section  of 
the  Brit.  Ass.  at  Exeter,  1869,  given   in  SYLVESTER'S  Laws  of  Verse, 
London,  1870,  p.  120.     This  eloquent  address  is  a  powerful  answer  to 
Huxley's   allegation    that    "Mathematics   is    that    study  which  knows 
nothing  of  observation,  nothing  of  experiment,  nothing  of  induction, 
nothing  of  causation."    We  quote  the  following  sentences  from  Sylves- 
ter :  "I,  of  course,  am  not  so  absurd  as  to  maintain  that  the  habit  of 
observation  of  external  nature  will  be  best  or  in  any  degree  cultivated 
by  the  study  of  mathematics."    "Most,  if  not  all,  of  the  great  ideas  of 
modern  mathematics  have  had  their  origin  in  observation."    "  Lagrange, 
than  whom  no  greater  authority  could  be   quoted,  has  expressed  em- 
phatically his  belief  in  the   importance   to  the   mathematician  of  the 
faculty  of  observation  ;  Gauss  called  mathematics  a  science  of  the  eye 
.  .  .  ;  the  ever  to  be  lamented  Kiemann  has  written  a  thesis  to  show 
that  the  basis  of  our  conception  of  space  is  purely  empirical,  and  our 
knowledge   of  its  laws  the  result  of  observation,  that  other  kinds   of 
space  might  be  conceived  to  exist  subject  to  laws  different  from  those 
which  govern  the  actual  space  in  which  we  are  immersed." 


284  A  HISTORY  OF   MATHEMATICS 

of  Thomas  Simpson's  geometry.  One  happy  event,  however, 
grew  out  of  these  discussions,  the  organization,  in  1870,  of 
the  Association  for  the  Improvement  of  Geometrical  Teaching 
(A.  I.  G.  T.).  T.  A.  Hirst,  its  first  president,  expressed  him- 
self as  follows:  "I  may  say  further,  that  I  know  no  suc- 
cessful teacher  who  will  not  admit  that  his  success  is 
almost  in  proportion  to  the  liberty  he  gives  himself  to 
depart  from  the  strict  line  of  Euclid's  Elements  and  to  give 
the  subject  a  life  which,  without  that  departure,  it  could 
not  possess.  I  know  no  geometer  who  has  read  Euclid  criti- 
cally, no  teacher  who  has  paid  attention  to  modes  of  exposi- 
tion, who  does  not  admit  that  Euclid's  Elements  are  full  of 
defects.  They  '  swarm  with  faults '  in  fact,  as  was  said  by 
the  eminent  professor  of  this  college  (De  Morgan),  who  has 
helped  to  train,  perhaps,  some  of  the  most  vigorous  thinkers 
of  our  time." l 

After  much  discussion  the  Association  published  in  1875  a 
Syllabus  of  Plane  Geometry,  corresponding  to  Books  I.— VI.  of 
Euclid.  It  was  the  aim  "  to  preserve  the  spirit  and  essentials 
of  style  of  Euclid's  Elements;  and,  while  sacrificing  nothing 
in  rigour  either  of  substance  or  form,  to  supply  acknowledged 
deficiences  and  remedy  many  minor  defects.  The  sequence  of 
propositions,  while  it  differs  considerably  from  that  of  Euclid, 
does  so  chiefly  by  bringing  the  propositions  closer  to  the  fun- 
damental axioms  on  which  they  are  based ;  and  thus  it  does 
not  conflict  with  Euclid's  sequence  in  the  sense  of  proving 
any  theorem  by  means  of  one  which  follows  it  in  Euclid's 
order,  though  in  many  cases  it  simplifies  Euclid's  proofs  by 
using  a  few  theorems  which  are  contained  in  the  sequence  of 
the  Syllabus,  but  are  not  explicitly  given  by  Euclid." 2  The 
Syllabus  received  the  careful  consideration  of  the  best  mathe- 

1  First  General  Report,  A.  I.  G.  T.,  1871,  p.  9. 

2  Thirteenth  General  Report,  1887,  pp.  22-23. 


ELEMENTARY   GEOMETRY  285 

matical  minds  in  England;  the  British  Association  for  the 
Advancement  of  Science  appointed  a  committee  to  examine 
the  Syllabus  and  to  co-operate  with  the  A.  I.  G.  T.  In  their 
report,  in  1877,  which  favoured  the  Syllabus,  was  expressed  the 
conviction  that  "no  text-book  that  has  yet  been  produced  is 
fit  to  succeed  Euclid  in  the  position  of  authority."  One  of 
the  great  drawbacks  to  reform  was  the  fact  that  the  great 
universities  of  Oxford  and  Cambridge  in  their  examinations 
for  admission  insisted  upon  a  rigid  adherence  to  the  proofs 
and  the  sequence  of  propositions  as  given  by  Euclid ;  thus  no 
freedom  was  given  to  teachers  to  deviate  from  Euclid  in  any 
way.  To  be  noted,  moreover,  was  the  total  absence  of  "  origi- 
nals "  or  "  riders,"  or  any  question  designed  to  determine  the 
real  knowledge  of  the  pupil.  But  in  the  last  few  years  the 
work  of  the  Association  has  been  recognized  by  the  universi- 
ties, and  some  freedom  has  been  granted.  It  was  noted  above 
that  J.  J.  Sylvester  was  an  enthusiastic  supporter  of  reform. 
The  difference  in  attitude  on  this  question  between  the  two  fore- 
most British  mathematicians,  J.  J.  Sylvester,  the  algebraist, 
and  Arthur  Cayley,  the  algebraist  and  geometer,  was  grotesque. 
Sylvester  wished  to  bury  Euclid  "  deeper  than  e'er  plummet 
sounded "  out  of  the  schoolboy's  reach ;  Cayley,  an  ardent 
admirer  of  Euclid,  desired  the  retention  of  Simson's  Euclid. 
When  reminded  that  this  treatise  was  a  mixture  of  Euclid  and 
Simson,  Cayley  suggested  striking  out  Simson's  additions  and 
keeping  strictly  to  the  original  treatise.1 

The  most  difficult  task  in  preparing  the  Syllabus  was  the 
treatment  of  proportion,  Euclid's  Book  V.  The  great  admira- 
tion for  this  Book  V.  has  been  inconsistent  with  the  practice 
in  the  school-room.  Says  Nixon,  "the  present  custom  of 
omitting  Book  V.,  though  quietly  assuming  such  of  its  results 

1  Fifteenth  General  Report,  A.  I.  G.  T.,  1889,  p.  21.  See  also  Four- 
teenth General  Report,  p.  28. 


286  A   HISTORY   OF   MATHEMATICS 

as  are  needed  in  Book  VI.,  is  singularly  illogical."1  Hirst 
offers  similar  testimony :  "  The  fifth  book  of  Euclid  .  .  .  has 
invariably  been  'skipped/  by  all  but  the  cleverest  school- 
boys."2 Here  we  have  the  extraordinary  spectacle  of  pro- 
portion relating  to  magnitudes  "  skipped,"  by  consent  of 
teachers  who  were  shocked  at  Legendre,  because  he  refers  the 
student  to  arithmetic  for  his  theory  of  proportion!  Is  not 
this  practice  of  English  teachers  a  tacit  acknowledgment  of 
the  truth  of  Raumer's 3  assertion  that  as  an  elementary  text- 
book the  Elements  should  be  rejected  ?  Euclid  himself  proba- 
bly never  intended  them  for  the  use  of  beginners.  But  the 
English  are  not  prepared  to  follow  the  example  of  other 
nations  and  cast  Euclid  aside.  The  British  mind  advances  by 
evolution,  not  by  revolution.  The  British  idea  is  to  revise, 
simplify,  and  enrich  the  text  of  Euclid. 

The  first  important  attempt  to  revise  and  simplify  the  fifth 
book  was  made  by  Augustus  De  Morgan.  He  published,  in 
1836,  "  The  Connexion  of  Number  and  Magnitude :  an  attempt 
to  explain  the  fifth  book  of  Euclid."  In  1837  it  appeared  as 
an  appendix  to  his  "Elements  of  Trigonometry."  It  is  on 
this  revision  that  the  substitute  for  Book  V.,  given  in  the 
Syllabus,  is  modelled.  On  the  treatment  of  proportion  there 
was  great  diversity  of  opinion  among  the  members  of  the  sub- 
committee appointed  by  the  Association  to  prepare  this  part 
of  the  Syllabus,  and  no  unanimous  agreement  was  reached.4 
There  were,  in  the  first  place,  different  schemes  for  re-arrang- 
ing and  simplifying  Euclid's  Book  V.  According  to  Euclid  four 
magnitudes,  a,  b,  c,  d,  are  in  proportion,  when  for  any  integers 

1  R.  C.  J.  NIXON,  Euclid  Revised,  Oxford,  1886,  p.  9. 

2  Fourth  General  Report,  A.  I.  G.  T.,  1874,  p.  18. 

8  Geschichte  der  Pddagogik,  Vol.  III. ;  see  also  K.  A.  SCHMID,  Encyklo- 
pddie  des  gesammten  Erziehungs-  und  Unterrichtswesens,  art.  "  Geometrie." 
4  NIXON,  op.  cit.,  p.  9. 


ELEMENTARY   GEOMETRY  287 

m  and  n,  we  have  simultaneously  ma  =  nb,  and  me  =  nd.     A 

definition  of  proportion  adopted  by  the  Italians  Sannia  and 
D'Ovidio  was  considered,  according  to  which  parts  are  sub- 
stituted for  multiples,  viz.  a,  b,  c,  d,  are  in  proportion,  if,  for 

any  integer  m,  a  contains  —  neither  a  greater  nor  a  less  number 

of  times  than  c  contains  —     Another  method  discussed  was 

m 

that  of  approaching  incommensurables  by  the  theory  of  limits, 
as  adopted  in  the  French  work  of  E.  Rouche  and  C.  de  Com- 
berousse,  and  by  many  American  writers.  This  method,  as  also 
Sannia  and  D'  Ovidio's  definition  of  proportion,  rests  on  the  law 
of  continuity  which  Clifford  defines  roughly  as  meaning  "  that 
all  quantities  can  be  divided  into  any  number  of  equal  parts." 1 
If  in  a  definition  it  is  desirable  to  assume  as  little  as  possible, 
Euclid's  definition  of  proportion  is  preferable,  inasmuch  as  it 
does  not  postulate  this  law.  To  minds  capable  of  grasping 
Euclid's  theory  of  proportion,  that  theory  excels  in  beauty 
the  treatment  of  incommensurables  with  aid  of  the  theory  of 
limits.  Says  Hankel,  "We  cannot  suppress  the  remark  that 
the  now  prevalent  treatment  of  irrational  magnitudes  in 
geometry  is  not  well  adapted  to  the  subject,  in  as  much  as  it 
separates  in  the  most  unnatural  manner  things  which  belong 
together,  and  forces  the  continuous  —  to  which,  from  its  very 
nature,  the  geometric  structure  belongs  —  into  the  shackles 
of  the  discrete,  which  nevertheless  it  every  moment  pulls 
asunder."  2 

About  the  time  of  Hankel  a  new  logical  difficulty  came  to 
light.  It  became  evident  that  the  theory  of  limits  as  it  ex- 
isted in  the  first  half  of  the  nineteenth  century  must  remain 
unsatisfactory  unless  a  theory  of  irrational  numbers  could 

1  The  Common  Sense  of  the  Exact  Sciences,  New  York,  1891,  p.  107. 

2  HANKEL,  op.  cit.,  p.  65. 


288  A   HISTORY   OF  MATHEMATICS 

be  presupposed.  As  it  was,  the  lines  of  procedure  involved 
reasoning  in  a  circle.  The  theory  of  limits  was  invoked  to 
establish  a  theory  of  irrational  numbers,  whereupon  the  theory 
of  irrational  numbers  seemed  to  be  used  for  a  more  complete 
exposition  of  the  theory  of  limits.  To  avoid  this  vicious  cir- 
cle, altogether  new  theories  of  irrational  numbers  were  devel- 
oped by  Karl  Weierstrass,  G.  Cantor,  and  R.  Dedekind,  which 
do  not  pre-require  the  theory  of  limits.1  These  developments 
have  greatly  strengthened  the  logical  foundations  of  mathe- 
matics, but  have  not  yet  been  put  in  a  form  suitable  for  use  in 
elementary  instruction.  From  the  standpoint  of  the  elementary 
teacher,  a  logical  exposition  of  irrational  numbers  and  of  limits, 
appears  impossible.  Neither  Euclid's  treatment  of  propor- 
tion, nor  the  treatment  of  irrational  number  and  limits  as 
given  by  modern  logicians,  is  proper  food  for  school  children. 
The  quest  for  perfect  logical  rigor  is  long  and  disheartening. 
Indeed  it  is  doubtful  whether  mathematicians  have  yet  reached 
absolute  rigor.  The  creations  of  G.  Cantor  and  Dedekind  have 
only  recently  given  rise  to  contradictory  results,  to  antinomies, 
that  are  thoroughly  disquieting.  To  be  sure,  the  introduction 
of  certain  restrictions  in  the  definitions  and  the  adoption  of 
certain  "axioms"  have  removed  the  antinomies  advanced 
thus  far.  But  we  know  not  the  day  nor  the  hour  when  new 
antinomies  will  spring  to  light.  In  view  of  the  subtlety  of 
the  modern  theories  of  numbers  and  of  doubt  as  to  their  real 
logical  perfection,  there  remains  no  alternative  for  the  ele- 
mentary teacher  but  to  reconcile  himself  with  a  logically 
imperfect  treatment  of  the  theory  of  limits  and  of  irrational 

1  Consult  GEORG  CANTOR,  Contributions  to  the  Founding  of  the  Theory 
of  Transfinite  Numbers,  translated,  and  provided  with  an  Introduction 
and  Notes,  by  Philip  E.  B.  Jourdain,  Chicago  and  London,  1915.  Jour- 
dain's  Introduction  traces  the  history  of  the  subject.  Consult  also  GEORGE 
BRUCE  HALSTED,  On  the  Foundation  of  Technic  of  Arithmetic,  Chicago, 
1912. 


ELEMENTARY   GEOMETRY  289 

numbers.1  In  fact  it  is  generally  felt  that  these  subjects  do 
not  really  belong  to  elementary  mathematics,  that  a  rudimen- 
tary and  purely  intuitive  treatment  of  limits  is  all  that  should 
be  attempted  in  high  school  and  in  the  first  stages  of  college 
mathematics. 

The  same  conclusion  on  matters  of  logic  is  reached  from  a 
different  angle.  Italian  mathematicians  under  the  leadership 
of  Giuseppe  Peano  of  the  University  of  Turin  set  themselves 
the  problem  to  adopt  definitions,  axioms,  and  undefined  terms 
so  that  geometry  might  be  built  up  without  appealing  to  intui- 
tion at  any  point.  As  is  well  known,  Euclid  quietly  resorts  to 
intuition  in  many  places,  for  the  knowledge  of  certain  facts. 
In  the  very  first  proposition  of  his  Elements  he  states  without 
proof  that  the  two  circles  which  he  draws  intersect  each  other. 
As  contributions  to  the  logical  exposition  of  geometry,  the  work 
of  the  Italian  school  was  praiseworthy.  A  culmination  of  this 
movement  is  found  in  a  treatise  prepared  by  the  great  German 
representative  of  logical  abstract  thought  in  mathematics, 
David  Hilbert  of  the  University  of  Gottingen.  At  the  unveil- 
ing of  the  Gauss-Weber  monument  in  Gottingen,  in  June,  1899, 
Hilbert  contributed  a  memorial  address  on  the  foundations  of 
Geometry,  entitled  Grundlagen  der  Geometric,  which  was  widely 
recognized  as  a  masterpiece  of  logical  exposition.  The  first 
edition  was  translated  into  English  in  1902  by  E.  J.  Townsend 
of  the  University  of  Illinois.  Several  German  editions,  some- 
what enlarged,  have  appeared  since.  Hilbert  sets  up  five 
groups  of  axioms,  which  include  altogether  twenty  axioms. 
To  show  how  Hilbert's  axioms  can  serve  as  a  foundation  of 
geometry,  built  up  without  appeal  to  intuition,  George  Bruce 
Halsted  published  in  1904  his  national  Geometry.  This  book 
affords  perhaps  the  easiest  approach  to  a  knowledge  of  Hilbert's 

1  For  an  illuminating  article  on  limits,  consult  BERTRAND  RUSSELL'S  arti- 
cle, "Mr.  Haldane  on  Infinity,"  in  Mind,  Vol.  33,  London,  1908,  p.  239. 


290  A   HISTORY   OF   MATHEMATICS 

system.  At  the  same  time  it  demonstrates  the  utter  impossi- 
bility of  introducing  high  school  students  into  geometry  by 
such  a  route.  To  meet  certain  logical  criticisms  of  his  text, 
Halsted  prepared  a  thoroughly  revised  second  edition  in  1907, 
which  has  been  translated  into  French. 


KECENT   MOVEMENTS   IN   TEACHING 

Toward  the  close  of  the  nineteenth  century  the  influence  of 
the  Association  for  the  Advancement  of  Geometric  Teaching 
(A.  I.  G.  T.),  whose  name  was  changed  in  1897  to  "  The  Mathe- 
matical Association  "  and  whose  work  had  been  broadened  in 
scope,  so  as  to  include  elementary  mathematical  instruction  in 
general,  showed  itself  in  many  ways.  To  note  the  progress 
in  geometry  we  need  only  examine  such  editions  and  revisions 
of  Euclid  as  those  of  Casey,  Nixon,  Mackay,  Langley  and 
Phillips,  Taylor,  and  the  Elements  of  Plane  Geometry  issued  by 
the  Association.1 

At  the  beginning  of  the  twentieth  century  there  were  two 
great  movements  in  the  teaching  of  elementary  mathematics. 
One  has  been  called  the  "  Perry  Movement,"  the  other  is  a  co- 
operative movement  of  teachers  of  different  nations  under  the 
name  of  the  "International  Commission  on  the  Teaching  of 
Mathematics."  Of  these  movements  the  first  made  at  once  a 
deep  and  lasting  impress,  especially  in  England  and  America, 
and  then  spent  itself ;  the  second  was  planned  on  a  large 
scale  and  was  still  in  the  preliminary  stages  when  interrupted 
by  the  great  war  which  broke  out  in  1914. 

1  For  the  history  of  geometrical  teaching  in  the  United  States  before 
1888,  see  CAJORI'S  The  Teaching  and  History  of  Mathematics  in  the  U.  S., 
1890. 


ELEMENTARY   GEOMETRY 


291 


The  Perry  Movement 

The  reform  which  this  movement  introduced  is  the  culmina- 
tion of  thirty  years  of  discussion  in  England,  carried  on  by  the 
members  of  the  Mathematical  Association.  This  culmination 
centers  around  the  personality  of  John  Perry,  then  Professor 
of  mechanics  and  mathematics  of  the  Royal  College  of  Science, 
London.  Perry  was  member  of  a  large  and  strong  committee 
of  the  British  Association  for  the  Advancement  of  Science, 
appointed  in  1901  to  report  on  the  "  Teaching  of  Mathematics." 
This  was  not  the  first  time  that  the  British  Association  had 
interested  itself  in  this  matter.  A  similar  committee  had  re- 
ported on  "  the  possibility  of  improving  the  methods  of 
instruction  in  elementary  geometry"  in  1873  and  again  in 
1876.1  These  early  reports  considered  the  advisability  of 
adopting  an  authorized  syllabus  of  geometry;  the  second 
report  considered  the  merits  of  the  syllabus  prepared  by  the 
A.  A.  G.  T.  and  recommended  it ;  nevertheless  that  syllabus 
had  not  been  generally  adopted  at  the  beginning  of  the  new 
century.  Teaching  continued  to  be  guided  mainly  by  the 
requirements  of  examinations. 

It  was  at  the  Glasgow  meeting  of  the  British  Association  in 
1901,  when  Section  A  (Mathematics  and  Physics)  and  Sec- 
tion L  (Education)  held  a  joint  meeting,  that  John  Perry 
delivered  a  great  address,  which  was  followed  by  a  brilliant 
discussion,  leading  to  epoch-making  results.  An  account  of 
these  deliberations  was  published  in  1901  under  the  title, 
Discussion  on  the  Teaching  of  Mathematics,  edited  by  John  Perry. 
This  debate  is  far  more  spicy,  interesting  and  valuable  than 
the  formal  report  of  the  committee,  made  a  year  later.  Perry 
called  for  a  complete  divorce  from  Euclid,  for  greater  emphasis 

1  See  Reports  of  the  British  Association  for  the  year  1873,  page  459, 
and  for  1876,  page  8. 


292  A   HISTORY   OF   MATHEMATICS 

upon  experimental  geometry,  for  practical  mensuration,  for 
the  use  of  squared  paper,  for  more  solid  geometry,  for  greater 
emphasis  upon  the  utilitarian  parts  of  geometry.  As  a  result, 
Euclid  was  dethroned  in  England  from  its  position  of  complete 
dominance  as  a  text-book  in  elementary  schools.  In  closing 
the  discussion,  Perry  said  : 

"  I  take  it  that  we  are  all  agreed  upon  the  following  points : 

"  1.  Experimental  methods  in  mensuration  and  geometry 
ought  to  precede  demonstrative  geometry,  but  even  in  the  ear- 
liest stages  some  deductive  reasoning  ought  to  be  introduced. 

"  2.  The  experimental  methods  adopted  may  greatly  be  left 
to  the  judgment  of  the  teacher :  they  may  include  all  those 
mentioned  in  the  Elementary  Syllabus  which  I  presented.  .  .  . 

"  3.  Decimals  ought  to  be  used  in  arithmetic  from  the  begin- 
ning. 

"4.  The  numerical  evaluation  of  complex  mathematical 
expressions  may  be  taken  up  almost  as  part  of  arithmetic  or 
at  the  beginning  of  the  study  of  algebra,  as  it  is  useful  in 
familiarizing  boys  with  the  meaning  of  mathematical  symbols. 

"  5.  Logarithms  may  be  used  in  numerical  calculation  as 
soon  as  a  boy  knows  an  x  am  =  am+n,  and  long  before  he  is  able 
to  calculate  logarithms.  But  a  boy  ought  to  have  a  clear 
notion  of  what  is  meant  by  the  logarithm  of  a  number. 

"  6.  In  mathematical  teaching  a  thoughtful  teacher  may  be 
encouraged  to  distinguish  what  is  essential  for  education  in 
the  sequence  which  he  employs,  from  what  is  merely  accord- 
ing to  arbitrary  fashion,  and  to  endeavor  to  find  out  what 
sequence  is  best,  educationally,  for  the  particular  kind  of  boy 
whom  he  has  to  teach. 

"  7.  Examination  cannot  be  done  away  with  in  England. 
Great  thoughtfulness  and  experience  are  necessary  qualifica- 
tions for  an  external  examiner.  It  ought  to  be  understood 
that  an  examination  of  a  good  teacher's  pupils  by  any  other 


ELEMENTARY   GEOMETRY  293 

examiner  than  the  teacher  himself  is  an  imperfect  examina- 
tion. .  .  . 

"  About  these  that  follow  I  am  in  more  doubt.  .  .  . 

"8.  A  thoughtful  teacher  ought  to  know  that  by  the  use 
of  squared  paper  and  easy  algebra,  by  illustrations  from 
dynamics  and  laboratory  experiments,  it  is  possible  to  give 
to  young  boys  the  notions  underlying  the  methods  of  the 
Infinitesimal  calculus. 

"9.  A  thoughtful  teacher  may  freely  use  the  ideas  and 
symbolism  of  the  calculus  in  teaching  elementary  mechanics 
to  students. 

"10.  A  thoughtful  teacher  may  allow  boys  to  begin  the 
formal  study  of  the  calculus  before  he  has  taken  up  advanced 
algebra  or  advanced  trigonometry,  or  the  formal  study  of 
analytical  or  geometrical  conies,  and  ought  to  be  encouraged  to 
use  in  this  study,  not  merely  geometrical  illustrations,  but 
illustrations  from  mechanics  and  physics,  and  illustrations 
from  any  other  quantitative  study  in  which  a  boy  may  be 
engaged." 

In  the  United  States  the  Perry  movement  did  not  bring 
about  as  many  changes  as  in  England.  There  are  two  prin- 
cipal reasons  for  this.  One  reason  is  that  Euclid  had  been 
discarded  long  ago  as  a  text  for  beginners,  hence  the  agitation 
in  Great  Britain  on  that  point  had  no  bearing  on  American 
conditions.  A  second  reason  is  that  some  of  the  other  recom- 
mendations of  Perry  did  not  directly  fit  the  American  situation. 
For  example,  Perry's  warfare  against  the  system  of  examina- 
tions then  in  vogue  in  British  universities  and  also  in  the 
English  civil  service  did  not  affect  America,  where  examina- 
tions were  conducted  differently  and  received  less  stress. 
The  clearest  American  declaration  with  respect  to  the  Perry 
movement,  in  so  far  as  it  affected  the  United  States,  came 
from  E.  H.  Moore  of  the  University  of  Chicago  in  his  presi- 


294  A   HISTORY   OF   MATHEMATICS 

dential  address,  "On  the  Foundations  of  Mathematics,"  de- 
livered on  December  29,  1902,  before  the  American  Mathe- 
matical Society.1  Moore  had  in  mind  mainly  college  mathe- 
matics, when  he  said :  "  In  agreement  with  Perry,  it  would 
seem  possible  that  the  student  might  be  brought  into  vital 
relation  with  the  fundamental  elements  of  trigonometry, 
analytic  geometry,  and  the  calculus,  on  condition  that  the 
whole  treatment  in  its  origin  is  and  in  its  development  remains 
closely  associated  with  thoroughly  concrete  phenomena." 
With  reference  to  secondary  schools,  Moore  puts  the  question : 
"  Would  it  not  be  possible  to  organize  the  algebra,  geometry, 
and  physics  of  the  secondary  school  into  a  thoroughly  coherent 
four  years'  course,  comparable  in  strength  and  closeness  of 
structure  with  the  four  years'  course  in  Latin  ? "  As  to 
method  of  instruction  :  "  This  program  of  reform  calls  for  the 
development  of  a  thoroughgoing  laboratory  system  of  instruc- 
tion in  mathematics  and  physics,  a  principal  purpose  being  as 
far  as  possible  to  develop  on  the  part  of  every  student  the 
true  spirit  of  research,  and  an  appreciation,  practical  as  well 
as  theoretic,  of  the  fundamental  methods  of  science." 

Experience  seems  to  show  that  the  success  of  the  laboratory 
method  of  mathematical  instruction  rests  upon  several  contin- 
gencies, such  as,  (1)  rearrangement  of  the  schedule  so  as  to 
allow  more  time  in  school  for  the  courses  in  mathematics, 
(2)  the  equipment  of  suitable  laboratories,  (3)  the  enlistment 
of  teachers  who  are  competent  to  give  laboratory  courses,  and 
so  on.  Even  if  all  these  conditions  prevail,  there  is  no  a  priori 
reason  why  the  laboratory  method  should  excel  other  methods. 
The  superiority  of  any  method  over  its  competitors  must  be 
determined  by  actual  trial.  Such  superiority  has  not  yet  been 
established  experimentally  for  the  laboratory  method. 

1  This  address  was  printed  in  Bulletin  of  the  Am.  Math.  Society,  Vol.  9, 
1903,  pp.  402-424,  also  in  Science,  N.  S.,  Vol.  17,  1903,  pp.  401-416. 


ELEMENTARY   GEOMETRY  295 

We  pass  to  the  broader  question,  the  unification  of  the  dif- 
ferent branches  of  high  school  mathematics,  carried  on  in  a 
manner  that  will  meet  more  effectively  the  practical  demands 
of  the  age.  This  question  has  been  under  discussion  and  ex- 
perimentation ever  since  the  opening  of  the  new  century.  The 
impression  seems  to  prevail  that,  in  past  decades  and  past  cen- 
turies, mathematics  never  were  unified,  that  teachers  placed 
mathematical  topics  into  separate  and  air-tight  compartments. 
Nothing  can  be  further  from  the  truth.  Previous  to  the  nine- 
teenth century  there  existed  a  fair  degree  of  fusion  of  different 
branches  of  mathematics.  We  see  this,  for  instance,  in  the 
Annies  papyrus,  in  Euclid's  Elements,  with  its  three  books  on 
arithmetic,  in  the  works  of  Vieta,  Stevin,  Oughtred,  Newton, 
Leibniz,  and  Euler. 

The  geometric  phraseology  "  linear  "  equation,  "  quadratic " 
equation,  "  cubic  "  equation,  "  a  square  "  for  a2,  "  a  cube  "  for 
a3,  shows  that,  in  its  formative  period,  algebra  was  not  disso- 
ciated from  geometry.  Moreover,  the  solution  of  cubic  and 
quartic  equations  by  the  method  of  geometric  construction  ap- 
pears earlier  in  the  history  of  mathematics  than  do  the  purely 
algebraic  solutions.  Isolation  in  mathematics  as  a  science  is  a 
product  of  the  nineteenth  century.  Steiner  and  Von  Staudt 
isolated  modern  geometry  from  analysis.  On  the  other  hand, 
Weierstrass,  Kronecker,  Klein,  and  others  worked  in  the  direc- 
tion of  arithmetization  which  makes  number  the  sole  founda- 
tion of  mathematics.  This  isolation  is  due  to  the  efforts 
toward  increased  rigor  of  presentation,  in  course  of  which  the 
foundations  of  mathematics  were  examined  anew.  Not  only 
in  the  growth  of  mathematics  as  a  science,  but  also  in  the 
teaching  of  mathematics,  isolation  of  topics  set  in.  Frequently, 
books  which  in  their  first  editions  exhibited  a  certain  degree  of 
fusion  of  topics,  passed  to  greater  isolation  in  later  editions. 
Thus,  Legendre's  Geometry,  first  published  in  1794,  stood  at 


296  A    HISTORY   OF   MATHEMATICS 

first  for  a  moderate  unification  of  geometry  with  arithmetic 
and  trigonometry.  As  late  as  1845  this  text  still  contained 
trigonometry,  but  later  the  trigonometry  and  the  practical 
applications  of  geometry  were  gradually  filtered  out.  Nicholas 
Pike's  New  and  Complete  System  of  Arithmetic,  printed  at  New- 
buryport,  Mass.,  in  1788,  and  used  for  a  time  even  in  our 
American  colleges,  contained  in  addition  to  the  ordinary  arith- 
metical topics,  a  treatment  of  logarithms,  trigonometry,  alge- 
bra, and  conic  sections.  The  fourth  edition  of  Pike's  book 
omitted  the  topics  not  strictly  arithmetical,  and  constituted 
Pike's  Abridged  Arithmetic.  Thus  there  arose  in  teaching  an 
isolation  of  topics,  a  moderate  degree  of  which  was  beneficial. 
Teachers  found  it  easier  to  reach  good  results  in  the  classroom 
by  taking  one  topic  at  a  time.  But  isolation  came  to  be  car- 
ried to  an  extreme.  The  exercises  of  the  schoolroom  related 
to  isolated  topics.  The  problems  of  practical  life  called  for 
ability  to  combine  arithmetic,  algebra,  and  geometry.  The 
pupils  lacked  experience  and  skill  in  combination.  Hence  the 
need  for  reunification.  No  agreement  has  yet  been  reached  as 
to  the  kind  and  degree  of  unification  that  is  most  suitable 
under  modern  conditions  of  industry  and  life.  There  is  dan- 
ger of  excessive  unification.  Too  much  fusion  may  produce 
confusion.  In  intense  fusion  there  is  danger  of  introducing 
simultaneously  a  large  number  of  new  ideas,  new  definitions. 
The  pupil  does  not  concentrate  upon  a  few  ideas,  but  scatters 
his  efforts  over  a  large  number  of  them  and  obtains  only  a 
superficial  view  of  any  one. 

If  it  be  asked  what  degree  of  unification  can  be  safely  adopted 
as  having  stood  the  test  of  experience,  the  answer  would  seem 
to  be  as  follows :  At  any  stage  of  mathematical  study,  let  the 
pupil  use  all  the  mathematics  that  he  has  had  up  to  that  time. 
For  instance,  when  he  studies  algebra,  let  him  use  freely  the 
arithmetic  that  preceded,  as  well  as  the  intuitive  geometry 


ELEMENTARY  GEOMETRY  297 

previously  learned  ;  when  pursuing  solid  geometry,  let  him  use 
the  arithmetic,  algebra,  and  plane  geometry  previously  studied. 
To  this  extent,  unification  is  sanctioned  by  experience.  How 
much  farther  we  can  go  advantageously  under  modern  condi- 
tions remains  to  be  determined  by  experience.  Doubtless 
some  allowance  must  be  made  for  individual  .predilections  of 
teachers  and  the  practical  needs  of  different  pupils. 

The  International  Commission 

At  the  Fourth  International  Congress  of  mathematicians 
that  was  held  at  Eome  in  1908,  David  Eugene  Smith  of  Teach- 
ers College,  Columbia  University,  made  a  motion  calling  for 
the  appointment  of  an  International  Commission  on  the  Teach- 
ing of  Mathematics.  The  motion  was  carried  and  a  Central 
Committee  was  appointed,  consisting  of  Felix  Klein  of  the 
University  of  Gottingen,  Sir  George  Greenhill  of  London,  and 
Henri  Fehr  of  the  University  of  Geneva  in  Switzerland.  In 
1912  David  Eugene  Smith  was  added  to  the  committee.  The 
Commission  adopted  as  its  official  organ  L' Enseignement  mathe- 
matique,  Revue  Internationale paraissant  tons  les  deux  mois.  This 
bi-monthly  journal  had  been  started  at  Paris  in  1899 ;  it  ap- 
pears under  the  editorship  of  C.  A.  Laisant  and  H.  Fehr.  Thus 
far  the  Commission  has  chiefly  occupied  itself  in  effecting 
organizations  in  the  different  countries,  and  in  collecting  and 
publishing,  for  comparative  examination,  the  fullest  possible 
information  as  to  the  present  status  of  the  teaching  of  mathe- 
matics in  each  of  the  countries  represented.  The  first  report 
of  the  International  Commission  was  made  at  the  Fifth  Inter- 
national Congress  of  Mathematicians,1  held  at  Cambridge,  Eng- 
land, in  1912.  At  that  time  there  were  27  countries  repre- 
sented, and  150  printed  reports  were  made,  leaving  about  50 

1  For  an  account  of  the  work  of  this  congress,  see  J.  W.  A.  YOUNG  in 
School  Science  and  Mathematics,  Vol.  12,  1912,  pp.  702-716. 


298  A    HISTORY    OF   MATHEMATICS 

still  to  be  published.  These  reports  contain  detailed  informa- 
tion on  the  condition  of  mathematical  teaching  at  different 
schools  and  in  different  countries.  They  are  important  contri- 
butions to  the  history  of  education.  Most  exhaustive  are  the 
numerous  reports  prepared  by  the  Germans.  The  British  re- 
ports are  models  of  concise  presentation.  The  documents  pre- 
pared in  the  United  States  by  special  committees  acting  under 
the  direction  of  the  American  Commissioners  (D.  E.  Smith  of 
Teachers  College,  W.  F.  Osgood  of  Harvard  University,  J.  W. 
A.  Young  of  the  University  of  Chicago)  have  been  published 
by  the  United  States  Bureau  of  Education.  The  breaking  out 
of  the  Great  War,  in  1914,  interrupted  the  work  of  the  Inter- 
national Commission. 

While  the  publications 1  hitherto  issued  are  mainly  historical, 
and  descriptive  of  present  conditions,  it  is  evident  that  the 
representatives  of  the  international  movement  stand  for  the 
more  thorough  embodiment,  in  mathematical  teaching,  of  cer- 
tain ideas,  at  least  a  part  of  which  were  emphasized  in  the 
Perry  movement.  Predominant  are  the  ideas  of  graphic  repre- 
sentation, functionality,  conservative  unification  of  mathe- 
matics, alignment  with  the  practical  demands  of  the  age.  In 
Germany  these  ideas  were  strongly  advocated  by  F.  Klein  at  a 
school  conference  held  in  1900.  In  France  these  same  ideas 
found  expression  in  the  text-books  published  by  E.  Borel  in 
1903,  which  were  translated  into  German  by  P.  Stackel  in 

1  The  French  reports  are  entitled,  Rapports  de  la  Commission  Interna- 
tionale de  rEnseignement  Mathematique,  Hachette  &  Company,  Paris. 
The  English  reports  are  brought  out  under  the  heading  The  Teaching  of 
Mathematics  in  the  United  Kingdom,  being  a  series  of  papers  prepared  for 
the  International  Commission  on  the  teaching  of  mathematics,  Wyman  & 
Sons,  Fetter  Lane,  London,  E.  C.  The  German  reports  are  Abhandlun- 
gen  ilber  den  mathematischen  Unterricht  in  Deutschland,  Veranlasst  durch 
die  Internationale  mathematische  Unterrichtskommission.  Herausgegeben 
von  F.  KLEIN,  Vols.  I-V,  B.  G.  Teubner,  Leipzic  u.  Berlin. 


ELEMENTARY   GEOMETRY  299 

1908,  to  serve  as  illustrations  of  what  can  be  done  in  the  way 
of  actual  realization  of  the  new  plans. 

The  subject  of  graphic  representation  of  mathematical  ex- 
pressions involving  variables  goes  back  to  the  time  of  Newton 
and  Descartes  ;  the  graphic  representation  of  curves  defined  by 
certain  properties  goes  back  to  the  time  of  the  Greeks. 

In  the  nineteenth  century  graphic  methods  came  to  be  used 
widely  in  mechanical  science,1  not  only  abroad  but  also  in  the 
United  States.  In  1898  E.  H.  Thurston  of  Cornell  University 
read  before  the  American  Society  of  Mechanical  Engineers  a 
paper  on  "Graphic  Diagrams  and  Glyptic  Models."  In  the 
teaching  of  elementary  mathematics  graphic  methods  were 
introduced  slowly.  Lone  voices  advocating  the  use  of  graphs 
in  teaching  were  heard  in  the  nineteenth  century. 

In  1842  Strehlke  of  Danzig  expressed  himself  as  follows:2 
"The  graphic  representation  of  functions  appears  to  me  an 
effective  aid  in  instruction,  for  the  purpose  of  conveying  a 
clear  conception  of  their  properties.  With  this  end  in  view 
I  have  drawn  a  considerable  number  of  such  graphs  —  the 
curves  of  sines,  cosines,  tangents,  of  natural  and  Briggian 
logarithms.  .  .  .  Even  simple  functions  like  y  =  x,  y  =  x2, 
y  =  x*,  y  =  x4,  if  they  are  drawn  for  the  same  variable  and 
upon  the  same  sheet,  present  properties  which  the  pupil  would 
not  see  the  moment  the  function  is  put  before  him."  In  the 
solution  of  equations,  graphic  methods  received  recognition  in 
L.  Saint-Loup's  Traite  de  la  resolution  des  equations  numtriques, 
Paris,  1861,  a  text  written  for  students  in  the  Polytechnic 
School  and  the  Normal  School  in  Paris,  "  a  Pusage  des  candi- 
tats  aux  ecoles  polytechnique  et  normale."  The  students  were 

1  See  the  "  Second  Report  on  the  Development  of  Graphic  Methods  in 
Mechanical  Science,"  by  H.  S.  HELE   SHAW,  in  Report  of  the  British 
Association  for  the  Advancement  of  Science  for  the  year  1892,  pp.  373-531. 

2  Grunert's  Archiv,  Vol.  2,  1842,  p.  111. 


300  A    HISTORY    OF   MATHEMATICS 

more  advanced  than  are  American  students  of  high  school 
grade.  In  the  Tutorial  Trigonometry,  by  William  Briggs  and 
G.  H.  Bryan,  London,  1897,  attention  is  given  to  the  tracing 
of  the  curves  of  the  trigonometric  functions. 

In  1896  appeared  Arthur  H.  Barker's  Graphical  Calculus, 
London,  which  represented  a  series  of  lectures  given  to  en- 
gineering students.  It  marks  a  complete  contrast  to  the 
ordinary  mathematical  book  of  the  school  of  Todhunter.  It 
is  a  work  intended  for  students  ready  to  enter  upon  the  study 
of  the  calculus. 

In  the  United  States  the  earliest  use  of  graphic  methods  in 
elementary  teaching  consisted  in  the  representation,  by  curves, 
of  empirical  data,  such  as  temperatures,  cloudiness  of  the  sky. 
From  1885  to  1889  such  methods  were  taught  in  the  school  of 
the  Chautauqua  Town  and  Country  Club,  to  about  two  thou- 
sand pupils.1  But  these  graphic  methods  met  with  no  general 
adoption  at  that  time.  In  1898  Francis  E.  Nipher  of  Wash- 
ington University  in  St.  Louis  brought  out  an  Introduction  to 
Graphic  Algebra.  The  first  regular  text-book  using  graphs  was 
the  Algebra  for  Schools  (1899)  by  George  W.  Evans  of  Boston. 
In  1902  there  appeared  as  a  supplement  to  Webster  Wells' 
Essentials  of  Algebra  and  New  Higher  Algebra,  a  monograph 
of  twenty-one  pages  on  Graphs,  prepared  by  Robert  J.  Aley, 
then  of  the  University  of  Indiana.  Henry  B.  Newson  of 
the  University  of  Kansas  followed  in  1905  with  a  Graphic 
Algebra  for  Secondary  Schools.  In  Vol.  5  (1905)  of  School 
Science  of  Mathematics  there  is  an  article  on  graphs  by 
F.  C.  Touton  of  Kansas  City.  Since  1905  our  algebras  for 
secondary  schools  introduce  the  graph  as  a  regular  subject  for 
instruction.  As  early  as  1903  "  the  use  of  the  graphic  methods 

1  See  CHARLES  BARNARD,  "Graphic  Methods  in  Teaching,"  Educa- 
tional Monographs  published  by  the  JV.  Y.  College  for  the  Training  of 
Teachers,  Vol.  II.,  No.  6,  1889,  pp.  209-223. 


ELEMENTARY   GEOMETRY  301 

and  illustrations,  particularly  with  the  solution  of  equations, 
is  also  expected  "  as  an  entrance  requirement  in  mathematics 
to  eastern  colleges. 

American  Associations 

There  have  been  organized  in  the  United  States  a  number 
of  associations  for  the  advancement  of  the  teaching  of  mathe- 
matics and  science  in  secondary  schools.  Thus  we  have  the 
Association  of  Teachers  of  Mathematics  in  New  England,  the 
Association  of  Ohio  Teachers  of  Mathematics  and  Science,  the 
Association  of  Teachers  of  Mathematics  in  the  Middle  States 
and  Maryland,  Central  Association  of  Science  and  Mathematics 
Teachers,  Colorado  Mathematics  Association,  Kansas  Associa- 
tion of  Mathematics,  Missouri  Society  of  Teachers  of  Mathe- 
matics and  Science,  Iowa  Association  of  Mathematics  Teachers, 
North  Dakota  Association  of  Science  and  Mathematics 
Teachers,  Southern  California  Science  and  Mathematics  Asso- 
ciation, Oregon  State  Science  and  Mathematical  Teachers 
Association. 

The  Mathematical  Association  of  America,  organized  in 
December,  1914,  devotes  itself  mainly  to  the  teaching  of  Math- 
ematics in  College.  The  American  Mathematical  Society  (or- 
ganized in  1888  as  the  New  York  Mathematical  Society,  and 
reorganized  under  its  present  name  in  1894)  devotes  itself 
entirely  to  the  furtherance  of  original  research. 

The  National  Educational  Association  (organized  in  Phila- 
delphia in  1857  under  the  name  of  "  National  Teachers'  Asso- 
ciation" and  re-named  in  1870)  has  always  interested  itself  in 
the  teaching  of  elementary  mathematics.  In  1893  a  "  Com- 
mittee of  Ten  "  reported  on  secondary  school  studies,  including 
the  recommendations  of  various  sub-committees  or  "confer 
ences."  A  "  conference  of  ten  "  on  mathematics  met  at  Har- 
vard in  December,  1892.  Simon  Newcomb  acted  as  chairman. 


302  A    HISTORY   OF   MATHEMATICS 

A  report  was  prepared,  which  is  given  in  outline  in  the  Report 
of  the  Commissioner  of  Education,  1892-1893,  Vol.  2,  p.  1426. 
Full  reports  of  the  "  conferences  "  were  published  separately 
in  1894.  It  was  recommended  by  the  mathematics  conference 
that  the  course  in  arithmetic  be  abridged  by  omitting  subjects 
which  perplex  and  exhaust  the  pupil  without  affording  mental 
discipline,  and  enriched  by  exercises  in  concrete  problems,  that 
geometry  be  first  taught  in  concrete  form,  including  measure- 
ments, and  that  greater  care  be  taken  to  avoid  slovenliness  of 
expression  in  demonstrations.  Stress  was  laid  upon  the  inter- 
lacing of  arithmetic,  algebra,  and  geometry.  The  eagerness 
with  which  the  various  reports  were  received  caused  a  "  Com- 
mittee of  Fifteen"  to  be  appointed  by  the  National  Educa- 
tion Association  to  institute  similar  inquiries  for  schools 
below  the  high  school.  Recommendations  were  made  on  the 
teaching  of  arithmetic.1  It  was  argued  that  the  simplest  frac- 
tions, i,  -J,  ^,  could  be  taught  very  early,  that  compound  in- 
terest, foreign  exchange,  and  certain  other  topics  be  dropped, 
that  mental  arithmetic  be  emphasized,  that  there  be  a  re- 
arrangement of  subjects,  and  that  the  rudiments  of  algebra  be 
introduced  in  the  seventh  year. 

In  1899  the  Committee  on  College  Entrance  Requirements, 
appointed  by  the  N.  E.  A.,  made  recommendations  in  mathe- 
matics which  were  in  practical  agreement  with  those  of  the 
Committee  of  Ten.  In  1904  the  Association  of  Mathematics 
Teachers  in  New  England  appointed  a  committee,  with  George 
W.  Evans  of  Boston  as  chairman,  to  consider  the  advisability 
of  preparing  a  geometry  syllabus,  or  adopting  some  syllabus 
already  drawn  up.  Such  a  syllabus  was  drawn  up ;  though 
carefully  prepared,  it  did  not  become  generally  known.  A 
report  by  a  syllabus  committee  of  the  Central  Association  of 
Science  and  Mathematics  Teachers,  made  in  1906,  attracted 

1  Report  of  the  U.  8.  Commissioner  of  Education  for  1893-1894,  p.  532. 


ELEMENTARY   GEOMETRY  303 

wider  attention,  but  was  imperfect  in  some  respects.  Finally, 
in  1908  and  1909  a  committee  was  organized  under  the  joint 
auspices  of  the  National  Education  Association  and  the 
American  Federation  of  Teachers  of  the  Mathematical  and 
National  Sciences.  H.  E.  Slaught  of  the  University  of  Chi- 
cago acted  as  chairman.  The  final  report  of  this  committee 
was  printed  in  1912,  and  received  much  attention  both  in  the 
United  States  and  abroad.  Theorems  are  classified  into  four 
groups,  according  to  their  importance,  the  groups  being  dis- 
tinguished one  from  the  other  by  the  kind  of  type  used.  The 
report  gives  considerable  attention  also  to  logical  considera- 
tions, and  contains  an  introduction  on  the  history  of  the  teach- 
ing of  geometry  during  the  last  two  centuries. 

In  recent  time  there  has  been  considerable  unrest  in  the 
United  States  on  matters  relating  to  mathematics  in  second- 
ary schools.  Proceeding  on  the  theory  that  the  real  aim  is 
not,  as  formerly  held,  to  "  train  the  mind,"  but  to  teach  only 
"  things  that  are  practical,"  so  that  the  mathematical  instruc- 
tion can  be  "  hitched  up  to  life,"  some  educators  exert  pressure 
to  omit  certain  subjects  formerly  thought  essential,  such  as 
the  treatment  of  factoring  and  fractions  in  algebra,  and  to 
restrict  problems  in  algebra  and  exercises  in  geometry  to  those 
that  are  practical.  In  some  ways  the  modern  tendency  re- 
sembles the  movement  of  the  fifteenth  and  sixteenth  centuries, 
led'  by  the  "  Commercial  School  of  Arithmeticians."  As  in  the 
sixteenth  century,  so  now,  there  is  meant  by  "  practical,"  not 
that  which  is  ultimately  practical,  but  that  which  is  immediately 
practical.  By  this  test  a  course  resembling  ancient  Egyptian 
geometry  makes  a  stronger  appeal  than  one  resembling  the 
more  scientific  Greek  geometry.  There  are  other  modern 
educators  who  go  farther  and,  denying  the  disciplinary  value 
of  mathematics,  insist  that  high  school  mathematics,  or  at  least 
a  part  of  it,  should  be  removed  from  the  required  list  of  high 


804  A   HISTORY    OF   MATHEMATICS 

school  studies.  It  may  be  of  interest,  in  this  place,  to  call 
attention  to  earlier  attacks  on  the  value  of  mathematics  as  an 
exercise  in  the  training  of  the  mind. 

Attacks  upon  the  Study  of  Mathematics  as  a  Training  of 
the  Mind.1 

Probably  the  most  famous  attack  that  has  ever  been  made 
upon  the  educational  value  of  mathematics  was  published  by 
the  philosopher,  Sir  William  Hamilton,  in  the  Edinburgh  Re- 
view of  1836.  He  claims  that  mathematics  is  "not  an  im- 
proving study " ;  "  If  we  consult  reason,  experience  and  the 
common  testimony  of  ancient  and  modern  times,  none  of  our 
intellectual  studies  tend  to  cultivate  a  smaller  number  of  the 
faculties,  in  a  more  partial  manner,  than  mathematics."  He 
proceeds  to  adduce  testimony  to  the  effect  that  "  the  cultiva- 
tion afforded  by  the  mathematics  is,  in  the  highest  degree, 
one-sided  and  contracted,"  that  mathematics  "  freeze  and  parch 
the  mind,"  that  this  science  is  "absolutely  pernicious  as  a 
mean  of  internal  culture."  Hamilton's  argument  proceeds 
along  two  principal  lines ;  the  first  is  to  prove  that  those  who 
confine  their  studies  to  mathematics  alone  are  addicted  to  blind 
credulity  or  irrational  scepticism  and,  in  general,  lack  good 
judgment  in  affairs  of  life.  We  admit  at  once,  that  this  last 
contention  is  probably  valid  ;  the  exclusive  study  of  any  branch 
of  knowledge  is  to  be  discouraged  as  undesirable  for  a  liberal 
education.  Hamilton's  second  mode  of  attack  consists  in  try- 
ing to  show  that  many  mathematicians  found  mathematics 
unsatisfactory  as  an  exercise  of  the  mind,  and  renounced  it. 
The  only  mathematicians  of  sufficient  prominence  to  be  known 

1  See  F.  CAJORI,  "  A  review  of  three  famous  attacks  upon  the  study  of 
mathematics  as  a  training  of  the  mind,"  in  the  Popular  Science  Monthly, 
1912,  pp.  360-372.  See  also  Dr.  C.  J.  KEYSER,  Mathematics,  Columbia 
Univ.  Press,  1907,  pp.  20-44. 


ELEMENTARY   GEOMETRY  305 

to  the  modern  reader  are  D'Alembert,  Pascal,  Descartes,  and 
Dugald  Stewart.  Many  other  men,  from  all  ages  and  nations, 
are  quoted,  and  constitute  indeed  a  "  cloud  of  witnesses."  Says 
Alfred  Pringsheim  of  Munich :  "  I  am  ashamed  to  confess  that 
before  reading  Hamilton's  article  I  did  not  know  a  single  one 
of  these  great  authorities  even  by  name ;  an  extenuating  cir- 
cumstance is  the  fact  that  some  of  these  names  I  could  not 
find  even  in  the  scientific  directories."  The  most  extensive 
examination  of  Hamilton's  quotations  and  most  thorough  dis- 
proof of  his  contentions  are  found  in  an  article  by  A.  T.  Bledsoe 
in  the  Southern  Review  for  July,  1877.  Bledsoe  was  at  one 
time  professor  of  mathematics  at  the  University  of  Virginia. 
He  proves  what  some  other  writers  before  him  hinted  at,  or 
proved  only  in  part,  namely,  that  Hamilton  was  extremely 
careless  in  the  selection  of  his  quotations.  By  making  partial 
extracts,  badly  chosen,  he  made  scientists  say  exactly  the  op- 
posite of  their  real  sentiments.1  Most  strikingly  is  this  true 
of  his  treatment  of  Descartes. 

The  second  great  attack  upon  mathematical  study  was  made 
by  Schopenhauer,  the  pessimistic  sage  of  Frankfort-on-the- 
Main.  In  his  The  World  as  Will  and  Idea,  second  edition, 
1844,  he  makes  war  on  Euclid  and  his  geometrical  demonstra- 
tions. "We  are  compelled  by  the  principle  of  contradiction 
to  admit  that  what  Euclid  demonstrates  is  true,  but  we  do  not 
comprehend  why  it  is  so.  We  have,  therefore,  almost  the 
same  uncomfortable  feeling  that  we  experience  after  a  juggling 
trick,  and,  in  fact,  most  of  Euclid's  demonstrations  are  re- 
markably like  such  feats.  The  truth  almost  always  enters  by 
the  back  door,  for  it  manifests  itself  par  accidens  through  some 
contingent  circumstance.  Often  a  reductto  ad  absurdum  shuts 
all  the  doors  one  after  another,  until  only  one  is  left,  which 

1  On  this  point  consult  also  JOHN  STUART  MILL'S  An  Examination  of 
Sir  William  Hamilton's  Philosophy,  New  York,  1884,  Vol.  II.,  pp.  305-336. 
P 


306  A- HISTORY    OF   MATHEMATICS 

we  are  therefore  compelled  to  enter."  This  is  part  of  Schopen- 
hauer's famous  characterization  of  mathematical  reasoning  as 
"mousetrap  proofs"  (Mausefallenbeweise).  His  objections 
are  directed  almost  entirely  against  Euclid.  He  had  no 
acquaintance  with  modern  mathematics.  His  criticism  of 
Euclid  as  a  text  for  children  is  largely  valid.  Euclid  did  not 
write  for  children.  It  is  a  historical  puzzle,  difficult  to  ex- 
plain, how  Euclid  ever  came  to  be  regarded  as  a  text  suitable 
for  the  first  introduction  into  geometry.  As  to  the  reductio  ad 
absurdum  method  of  proof.  De  Morgan  said :  "  The  most 
serious  embarrassment  in  the  purely  reasoning  part  is  the 
reductio  ad  absurdum,  or  indirect  demonstration.  This  form 
of  argument  is  generally  the  last  to  be  clearly  understood, 
though  it  occurs  almost  on  the  threshold  of  the  elements" 
Schopenhauer's  attack  bears  only  indirectly  upon  the  question 
relating  to  the  mind-training  value  of  mathematics.  His.  views 
must  have  attracted  considerable  attention  in  Germany,  for  as 
late  as  1894  A.  Pringsheim  thought  it  necessary  to  refute  his 
argument,  and  only  nine  years  ago  Felix  Klein  referred  to 
him  at  some  length  in  a  mathematical  lecture  at  the  University 
of  Gottingen. 

A  third  attack  upon  mathematics  was  made  in  1869  by  the 
naturalist,  Thomas  H.  Huxley.  In  the  Fortnightly  Eeview, 
1869,  p.  433,  he  said :  Mathematics  is  that  study  "  which 
knows  nothing  of  observation,  nothing  of  experiment,  nothing 
of  induction,  nothing  of  causation."  In  an  after-dinner  speech, 
reported  in  Macmillan's  Magazine,  1869,  pp.  177-184,  he  argued 
against  language  and  mathematical  training,  and  in  favor  of 
scientific  training.  "Mathematical  training,"  he  said,  "is 
almost  purely  deductive.  The  mathematician  starts  with  a 
few  propositions,  the  proof  of  which  is  so  obvious  that  they 
are  called  self-evident,  and  the  rest  of  his  work  consists  of 
subtle  deductions  from  them."  Of  great  interest  is  the  reply 


ELEMENTARY   GEOMETRY  307 

made  to  Huxley  by  the  mathematician,  J.  J.  Sylvester.1  To 
Americans  Sylvester's  name  is  memorable,  because  in  1876 
Sylvester  came  over  from  England  and  for  eight  years  taught 
modern  higher  algebra  to  American  students  at  the  Johns 
Hopkins  University  and  gave  a  powerful  stimulus  to  the  study 
of  higher  mathematics  in  this  country.  Sylvester  was  an 
enthusiast.  His  reply  to  Huxley  was  his  presidential  address 
before  Section  A  of  the  British  Association  in  1869.  By  his 
own  experiences  as  a  mathematician  he  tried  to  show  that 
Huxley  was  wrong.  Sylvester  insists  "  that  mathematical 
analysis  is  constantly  invoking  the  aid  of  new  principles,  new 
ideas  and  new  methods  .  .  .  springing  direct  from  the  inher- 
ent powers  and  activity  of  the  human  mind  .  .  .  that  it  is 
unceasingly  calling  forth  the  faculties  of  observation  and  com- 
parison, that  one  of  its  principal  weapons  is  induction,  that  it 
has  frequent  recourse  to  experimental  trial  and  verification, 
and  that  it  affords  a  boundless  scope  for  the  exercise  of  the 
highest  efforts  of  imagination  and  invention.  Lagrange  .  .  . 
expressed  emphatically  his  belief  in  the  importance  to  the 
mathematician  of  the  faculty  of  observation ;  Gauss  called 
mathematics  the  science  of  the  eye  ...  I  could  tell  a  story  of 
almost  romantic  interest  about  my  own  latest  researches  in  a 
field  where  geometry,  algebra,  and  the  theory  of  numbers  melt 
in  a  surprising  manner  into  one  another,  .  .  .  which  would 
very  strikingly  illustrate  how  much  observation,  divination, 
induction,  experimental  trial,  and  verification,  causation  too, 
.  .  .  have  to  do  with  the  work  of  the  mathematician."  Syl- 
vester proved  conclusively  that  the  mathematician  engaged  in 
original  research  does  exercise  powers  of  internal  observation, 
induction,  experimentation,  and  even  causation.  The  vital 
question  arises,  are  these  powers  exercised  by  the  pupil  in  the 
classroom?  That  depends.  If  a  text-book  in  geometry  is 
1  See  page  283,  last  footnote. 


308  A    HISTORY   OF    MATHEMATICS 

memorized,  as  used  to  be  the  case  in  some  schools,  then  doubt- 
less Huxley  was  right.  But  when  attention  is  paid  to  original 
problems,  to  the  use  of  measuring  instruments,  to  modes  of 
proving  theorems  different  from  those  in  the  book,  then  the 
pupil  exercises  the  same  faculties  as  does  the  advanced  mathe- 
matician engaged  in  research. 

Perhaps  the  latest  prominent  attacks  on  mathematics  are  by 
David  Snedden,  Commissioner  of  Education  in  Massachusetts, 
in  an  address  delivered  in  1915  before  the  National  Educa- 
tion Association,  and  by  Abraham  Flexner  in  an  article, 
called  "  A  Modern  School." l  Placing  great  faith  in  the 
methods  of  experimental  pedagogy  that  are  advocated  in 
schools  of  education  of  our  universities  at  the  present  time, 
these  writers  are  attacking  the  mind-training  value  of  mathe- 
matics and  are  insisting  that,  in  view  of  the  results  obtained 
from  experiments,  the  burden  of  proof  to  show  that  algebra 
and  geometry  should  remain  in  the  high  school  curriculum 
rests  upon  those  who  advocate  its  continuance  and  not  upon 
those  who  would  eliminate  them.  Says  Flexner :  "  Modern 
education  will  include  nothing  simply  because  tradition  recom- 
mends it,  ...  it  includes  nothing  for  which  an  affirmative 
cannot  now  be  made  out.  As  has  already  been  intimated, 
this  method  of  approach  would  probably  result  in  greatly  re- 
ducing the  time  allowed  to  mathematics,  and  decidedly  chang- 
ing the  form  of  what  is  still  retained.  If,  for  example,  only 
so  much  arithmetic  is  taught  as  people  actually  have  occasion 
to  use,  the  subject  will  sink  to  modest  proportions.  .  .  .  The 
same  policy  may  be  employed  in  dealing  with  algebra  and 
geometry.  What  is  taught,  when  it  is  taught,  and  how  it  is 
taught  will  in  that  event  depend  altogether  on  what  is  needed, 

1  Beview  of  Beviews,  Vol.  53,  1916,  pp.  466-474.  See  also.  A. 
FLEXNER'S  "Parents  and  Schools,'*  in  the  Atlantic  Monthly,  July,  1916, 
pp.  25-33. 


ELEMENTARY   GEOMETRY  309 

when  it  is  needed,  and  the  form  in  which  it  is  needed."  It 
will  be  instructive  to  watch  the  outcome  of  this  discussion. 
Certain  it  is  that  the  teaching  of  mathematics  will  require 
constant  modifications  in  order  to  keep  pace  with  the  ever- 
changing  needs  of  modern  life. 


INDEX 


Abacists,  115, 118,  119,  188. 

Abacus,  11,  14,  17,  26-28,  37-41,  106, 

112,  114, 117-119,  186. 
Abel,  227,  238. 

Absolute  geometry,  273.     See  Non- 
Euclidean  geometry. 
Abu  Dscha  'far  Alchazin,  110. 
Abu  Ja'kub  Ishak  ibn  Hunain,  127. 
Abu'l  Dschud,  110. 
Abul  Gud.    See  Abu'l  Dschud. 
Abu'l  Wafa,  104, 107,  128,  130,  248. 
Achilles  and  tortoise,  paradox  of,  59. 
Acre,  178, 179. 
Aaalbold,  132. 
Adams,  D.,  218,  219,  222. 
Addition,  26,  37,  38,  96,  101, 105,  110, 

115,  145.    See  Computation. 
Adelhard  of  Bath.    See  Athelard  of 

Bath. 

Aeneas,  172. 
Agrimensores,  89-92. 
Ahmes,   19-25,   28,  34,  43-45,  47,  82, 

120,  222. 
A.  I.  G.  T.,  68,  69,  206,  208,  213,  263, 

268,  279,  284-286,  288. 
Akhmim  papyrus,  25,  82. 
Albanna,  ibn,  106,  111,  150. 
Al  Battam,  118,  130. 
Albategnius.    See  Al  Battani. 
Albiruni,  15,  104. 
Alchwarizmi,  104-109,  118,  128,  129, 

133,  206. 

Alcuin,  112-114,  119,  131,  220. 
Aldschebr  walmukabala,  107, 108. 
Alexandrian    School,    First,    64-82; 

Second,  82-89. 
Algebra,  in  Egypt,  23-25 ;  in  Greece, 

26,  33-37;   in  Rome,  42;  in  India, 


93-95,  101-103 ;  in  Arabia,  104, 106- 

111;    Middle  Ages,    118,  120,  122, 

133;   Modern  Times,  139,  141,156, 

183,  206,    209,    224-245,    283,    288; 

origin  of  word,  107.    See  Notation. 
Algorism,  119;  origin  of  word,  105. 
Algoristic  school,  118, 119,  188. 
Algorithm.    See  Algorism. 
Al  Hovarezmi.     See  Alchwarizmi. 
Aliquot  parts,  23,  197,  218. 
Alkalsadi,  110,  150. 
Al  Karchi,  106,  110. 
Alligation,  110,  185,  198. 
Allman,  47,  48,  50,  51,  53,  56,  57,  59, 

62,  63,  66. 
Al  Mahani,  110. 

Al  Mainun,  126, 128.     Almansur,  129. 
Alnager,  176. 
Alsted,  203. 

Amicable  numbers,  29. 
Amyclas,  63. 
Analysis,  in  ancient  geometry,  61,62; 

in  arithmetic,  196,  197;  algebraic, 

249. 

Anaxagoras,  48,  49,  53. 
Anaximander,  48. 
Anaximenes,  48. 
Angle,  trisection  of,  34,  55,  134,  135, 

226. 

Anharmonic  ratio,  258. 
Annuities,  198. 

Anthology,  Palatine,  33,  34,  113. 
Anti-parallel,  262. 
Antiphon,  57-59. 
Apian,  141. 
Apices  of  Boethius,  12, 14-16, 112, 114, 

115,  118. 
Apollonian  Problem,  79. 


311 


312 


INDEX 


Apollonius,  65,  75,  78-80,  86,  87,  104, 
127,  247,  250,  255. 

Apothecaries'  weight,  173. 

Appuleius,  32. 

Arabic  notation .    See  Hindu  notation . 

Arabs,  10,  13-18,  24,  65,  84,  95,  97, 
103-111,  116,  118,  119,  124-132,  134, 
137,  147,  150,  188,  224,  238. 

Archimedes,  28,  33,  63-65,  73,  75-79, 
86,  127,  134,  158,  242,  250,  254. 

Archytas,  53,  60,  62,  63. 

Arenarius,  29. 

Argand,  243. 

Aristotle,  29,  47,  57,  61,  76, 132,  275. 

Arithmetic,  in  Egypt,  19-26,  82;  in 
Greece,  26-37;  in  Rome,  37-42;  in 
India,  93-101;  in  Arabia,  103-111; 
in  Middle  Ages,  111-122 ;  in  Modern 
Times,  139-221, 145-168 ;  in  England, 
20,  168,  179-211 ;  Reforms  in  teach- 
ing, 211-219 ;  in  the  United  States, 
215-223.  See  Computation,  Hindu 
notation,  Notation,  Number-sys- 
tems. 

Arithmetical  triangle,  238. 

Arneth,  125. 

Arnold,  M.,  207. 

Arnold,  T.,  207. 

Artificial  numbers.    See  Logarithms. 

Aryabhatta,  12,  94,  99,  100, 102,  123. 

Assumption,  tentative.  See  False 
position. 

Athelard  of  Bath,  104,  118,  132, 133, 
136. 

Athenaeus,  63. 

Atkinson,  275. 

Attic  signs,  7. 

Austrian  method,  of  subtraction,  213; 
of  division,  214,  215. 

Avoirdupois  weight,  173, 174, 196, 198. 

Axioms,  46,  61,  67,  69-71,  89,  90,  275, 
276,  279,  281.  See  Postulates,  Par- 
allel-postulate. 

Babylonians,  1,5,  6,  8,  9, 10,20, 23,  30, 

43,  44,  49,  79,  84,  88,  123,  175. 
Bache,  215. 

Bachet  de  Meziriac,  220,  222. 
Bachmann,  265. 
Backer  rule  of  three,  198. 
Bacon,  R.,  137. 


Baillet,  25. 

Baker,  187,  193. 

Bakhshali  arithmetic,  94,  98,  99. 

Ball,  W.  W.  R.,  16,  13:!,  180, 184,  206, 

242,  281. 

Baltzer,  R.,  273,  280. 
Bamberg  arithmetic,  16,  140, 180. 
Barley-corn,  175, 177. 
Barreme,  192. 
Barter,  198. 

Base,  logarithmic,  160. 
Basedow,  197. 
Battaglini,  278. 
Bayle,  246. 
Bede,  112,  119. 
Beha  Eddin,  128.      , 
Bekker,  57. 
Benese,  R.  de,  178. 
Berkeley,  144. 
Bernelinus,  114,  115. 
Bernoulli,  James,  237. 
Bernoulli,  John,  200. 
Be  van,  B.,  259. 
Beyer,  153. 
Be-zout,  267,  276. 
Bhaskara,  50,  95,  99,   101,  102,  103 

123,  205. 

Billingsley,  246,  281. 
Billion,  144. 
Binomial  surds,  103. 
Binomial  theorem,  209,  229,  237,  238; 

not  inscribed  on  Newton's  tomb, 

239. 

Biot,  257. 

Biquadratic  equations,  226. 
Bitonto,  267. 
Bockler,  154. 
Boethius,  12,  14,  15,  16,  32,  41,  91,  92, 

111,  112,  114,  115,  117,  118,  131,  132, 

133,  136,  142,  147,  188. 
Bolyai,  J.,  70,  71,  86,  127,  268,  269, 

273, 274,  275,  279. 
Bolyai,  W.,  273. 
Bombelli,  226,  227. 
Boncompagni,  105, 142. 
Bonnycastle,  244,  267. 
Bookkeeping,  121, 168, 188,  206. 
Borelli,  268. 
Boscovich,  266. 
Bouelles.    See  Bouvelles. 
Bourdon,  208. 


INDEX 


313 


Bouvelles,  247. 

Bovillus,  247. 

Brackett,  208. 

Bradwardine,  136, 137,  247,  250. 

Brahe,  Tycho,  250. 

Brahmagupta,  94,  95,  100,  101,  102, 

129,  196,  205. 

Bretschneider,  47,  50,  56,  62,  123. 
Brewer,  172. 
Brewster,  239,  277,  283. 
Brianchon,  248,  252,  253,  257,  260. 
Brianchon  point,  257. 
Brianchon's  theorem,  257. 
Bridge,  244. 
Bridges,  188. 
Briggian  logarithms,  159,  162,   165, 

199, 

Briggs,  161-163,  167,  179,  238. 
Briot  and  Bouquet,  227. 
Brocard,  H.,  259,  261. 
Brocard  circle,  261,  262. 
Brocard  points  and  angles,  261. 
Brocard's  first  and  second  triangle, 

261. 

Brockhaus,  159. 
Brocki,  J.    See  Broscius. 
Brown,  200. 
Broscius,  247. 
Brute,  172. 

Bryson  of  Heraclea,  58,  59. 
Buckley,  150,  187,  188, 189. 
Buddha,  29. 
Buee,  243. 
Burgess,  12. 
Biirgi,  153, 166. 
Burkhardt,  226. 
Bushel,  172. 
Butterworth,  260. 
Byllion,  144. 
Byrgius.    See  Biirgi. 

Caesar,  Julius,  90,  91. 

Cajori,  26,  29,  159,  237,  238,  304. 

Campano.    See  Campanus. 

Campanus,  134,  135, 136,  245,  247. 

Campbell,  T.,  137. 

Cantor,  G.,  70. 

Cantor,  M.,  5,  6,  7,  8,  10,  12,  14, 16, 
19-26,  31,  34,  38,  43-45,  47,  50,  56, 
59,  60,  67,  76,  78,  80,  81,  83-85,  87, 
90,  94,  98-102, 105, 110-113, 118, 120, 


122, 127, 129, 130,  132,  133, 134,  136, 
137,  141-145,  150,  181,  184,  194,  222, 

227,  231,  232,  235,  238,  245,  247-249, 
250,  252,  282. 

Capella,  91,  111,  131. 

Cardan,  150,  151,  222,  224,  225,  227, 

228,  229,  230,  232,  234,  240. 
Cardano.    See  Cardan. 
Carnot,  L.,  83,  257,  369. 
Casey,  259,  288. 
Cassiodorius,  91,  92,  111,  112. 
Castellucio,  146. 

Casting  out  the  9's,  98, 106,  149,  184, 

185. 

Cataldi,  150, 187. 
Cattle-problem,  33. 
Cauchy,  244. 
Cavalieri,  209. 
Caxton,  16, 180. 
Cayley,  285. 
Ceva,  83,  256. 
Ceva's  theorem,  256. 
Chain,  179. 
Chain-rule,  196, 197. 
Champollion,  6. 
Charlemagne,  168,  220. 
Charles  XII.,  2,  205. 
Chasles,  74,  83,  87,  89,  248,  252,  255, 

256,  258,  259. 
Chauvenet,  278. 
Chebichev,  31. 
Chinese,  94,  123,  240. 
Chuquet,  141,  142,  144,  222. 
Cicero,  75. 

Cipher,  152;  origin  of  word,  119. 
Circle,  48,  52,  53,  54,  57,  58,  59,  73,  75, 

78, 79,  87, 250, 255,  265, 276 ;  division 

of,  10,  43,  73,  74,  79,  84, 124, 134,  265. 

See  Squaring  of  the  circle. 
Circle-squaring.    See  Squaring  of  the 

circle. 

Circulating  decimals,  200. 
Clairaut,  269,  276,  277. 
Clausen,  Th.,  242. 
Clavius,  71,  137, 176,  266. 
Clifford,  W.  K.,  71,  258,  275,  287. 
Cloff,  198. 
Cocker,  31,  173, 184,  190, 191, 192-194, 

196, 199,  202,  203,  210,  212,  217. 
Colburn,  W.,  218,  219. 
Colebrook,  96. 


314 


INDEX 


Colla,  224. 

Combinations,  doctrine  of,  237. 

Commercial  School  in  England,  179- 

204. 
Common   logarithms,   159,   162,  165, 

199. 

Common  notions,  70. 
Compasses,  single  opening    of,   128, 

248,  264,  265.     See  Ruler  and  corn- 


Complementary    division,    116,    117, 
147;  —multiplication,  147. 

Complex  numbers,  244.     See  Imag- 
inaries. 

Compound  interest,  151.   See  Interest. 

Computation,  modes  of,  in  Ahmes, 
21-24  ;  in  Greece,  27,  28,  32,  37-42 ; 
in  India,  95-100;  in  Arabia,  104 
106;  Middle  Ages,  111,  112, 114-122; 
Modern  Times,  145-167,  186,  187, 
206 ;  in  England,  205,  206,  241 ;  in 
the  United  States,  241. 

Computus,  112. 

Comte,  v. 

Conant,  1,  3,  4,  5. 

Conic  sections,  63,  78,  254,  256. 

Conjoined    proportion.     See   Chain- 
rule. 

Continued  fractions,  150,  200. 

Continuity,  law  of,  in  algebra,  237, 
287  ;  in  geometry,  237,  252,  268,  287. 

Cooley,  267. 

Cosine,  124,  162. 

Cossic  art,  228.    See  Algebra. 

Cotangent,  130,  137,  163. 

Cotes,  208,  256. 

Counters,  115,  118,  185-187. 

Crelle,  227,  238,  260,  261. 

Cremona,  248,  278,  279. 

Cridhara,  95,  102. 

Cross-ratio,  255,  258. 

Ctesibius,  80. 

Cube,  duplication  of,  53,  54,  56,  226. 

Cube  root,  101,  106,  150,  228,  241. 

Cubic  equations,  110, 120, 129, 224-226, 
227,  229,  230,  240. 

Cubical  numbers,  32. 

Cubit,  175. 

Cunn,  55,  200. 

Curtz,  134,  222. 

Cyzicenus,  63. 


Daboll,  218. 

Damascius,  67,  88,  127. 

Dase,  166,  242. 

Data  of  Euclid,  74,  133,  246. 

Davies,  189. 

Davies,  C.,  277. 

Davies,  F.  S.,  260. 

Da  Vinci,  Leonardo,  141,  248,  249. 

De  Benese,  178. 

Decimal  fractions,  150-155,  166,  187, 

199,  200,  214. 

Decimal  point,  153,  154,  194,  195. 
Decimal  scale,  1,  2,  3,  4,  5,  8,  11,  40, 

85,  152,  163,  165,  166,  178. 
Dedekind,  72. 
Dee,  John,  246. 
Defective  numbers,  29,  113. 
Degrees,  10,  179. 
De  Jonquieres,  265. 
De  Lagny,  241,  242. 
De  Lahire,  252,  253. 
De  la  Hire.    See  De  Lahire. 
Delbosuf ,  269. 
Del  Ferro.    See  Ferro. 
Delian  Problem,  55.     See  Duplication 

of  the  cube. 
Democritus,  45,  46. 
De  Morgan,  55,  64,  65,  68,  72, 122, 141, 

151,  154,  164,  175,  178,  181,  187-190, 

199,  200,  205,  208,  210,  213,  238,  240, 

241,  244,  246,  268,  283,  284,  286. 
Demotic  writing,  7. 
Desargues,  237,  252,  253,  255,  257,  269. 
Desargues'  Theorem,  253. 
Descartes,  233,  234,  235,  236,  238,  243, 

252,  254,  256,  265. 
Descartes'  Rule  of  Signs,  236. 
Devanagari-numerals,  14,  16,  17. 
Dilworth,  160,  174,  191,  194,  195,  198, 

201,  202,  203,  217,  218,  219,  220. 
Dinostratus,  63. 
Diogenes  Laertius,  28,  47,  61. 
Diophantine  analysis,  37,  101. 
Diophantus,  26,  34-37,  82,  87, 101, 102, 

104,  108,  109,  127. 
Direction,  concept  of,   in  geometry, 

268,  280,  289. 
Dirichlet,  31. 
Divisio  aurea,  117. 
Divisio  ferrea,  117. 
Division,  37,  39,  98,  101,  105,  114, 115- 


INDEX 


315 


118,  140,  148,  149,  150,  151,  154,  156, 

183,  189,  192,  213,  214. 
Dixon,  E.  T.,  68,  268. 
Dodecaedron ,  51 .     See  Regular  solids. 
Dodgson,  267,  268,  270. 
Dodson,  195,  199. 
Dollar  mark,  223. 
Double  position,  106,   196,   198,  203. 

See  False  position. 
Dschabir  ibn  Aflah,  107,  129,  130. 
Duality,  258. 
Diicange,  143. 
Duhamel,  77,  278. 
Duodecimal  fractions,  38-41,  119. 
Duodecimal  scale,  2,  3,  112,  117. 
Dupin,  257. 
Duplication  of  the  cube,  53,  54,  56, 

226. 

Durege,  266. 
Diirer,  221,  248,  249,  266. 

Easter,  computation  of  time  of,  40, 

112.     See  Computus. 
Egyptians,  1,  6,  7,  19-26,  27,  44-46, 

49,  52,  55,  76,  88-90,  99,  122,  131, 

170,  175. 
Eisenlohr,  19. 
Elefuga,  137. 
Elements  of    Euclid.      See    Euclid's 

Elements. 

Emmerich,  259,  262. 
EnestrSm,  141. 
Engel  and  Stackel,  68,  70,  72,  267, 

269-272,  274. 
Epicureans,  74. 

Equality,  symbol  for,  184,  195. 
Equations,  linear,  23,  33,  37,  107,  108 ; 

quadratic,  36, 101, 102, 108, 110,  230; 

cubic,   110,  120,  129,  224-227,   229, 

240;  simultaneous,  35;  higher,  226, 

227,230,239;  numerical,  240;  theory 

of,  230-233,  236,  243. 
Eratosthenes,  31,  56,  65. 
Eton,  204,  205,  209. 
Euclid,  31,  35,  46,  47,  50,  52,  60,  62,  63, 

64-75,  76,  77,  79,  80,  83,  85,  86,  104, 

134,  135,  272,  274,  276,  277,  278. 
Euclid  of  Megara,  65,  246. 
Euclid's  Elements,  30,  65-75,  79,  81, 

84,  87,  88,  91,  103,  126,  127,  133,  136- 

138,  206,  207,  212,  238,  245,  246,  254, 


270,  275-289;  editions  of  the  Ele- 
ments, 245,  246,  266,  282,  288. 

Euclid  and  his  modern  rivals,  267, 268, 
275-289. 

Eudemian  Summary,  47-49,  61,  63, 64. 

Eudemus,  47,  52. 

Eudoxus,  46,  60,  62,  63,  64,  66. 

Euler,  36,  157,  195,  200,  238,  243,  259, 
265. 

Eutocius,  26. 

Evolution.  See  Square  root,  Cube 
root,  Roots. 

Excessive  numbers,  29,  111. 

Exchange,  121,  198. 

Exchequer,  174,  186,  215. 

Exhaustion,  method  of,  59,  60,  62,  63, 
66. 

Exhaustion,  process  of,  57,  59. 

Exponents,  122,  152, 183,  234-238. 

Fallacies,  74. 

False  position,  24,  36, 99, 106, 141, 184, 
185,  196,  198,  240. 

Farrar,  J.,  243,  277. 

Farthing,  derivation  of  word,  170. 

Fellowship,  121,  185. 

Penning,  191,  217. 

Fermat,  209,  252. 

Ferrari,  225,  226,  228. 

Ferreus.    See  Ferro. 

Ferro,  224,  225. 

Feuerbach,  260. 

Feuerbach  circle,  259,  260.  See  Nine- 
point  circle. 

Fibonacci.    See  Leonardo  of  Pisa. 

Figure  of  the  Bride,  128. 

Finasus,  147,  150,  151. 

Finger-reckoning,  1,  2,  17,  27,  40,  112. 

Fiore,  Antonio  Maria.    See  Floridus. 

Fisher,  George,  217,  192. 

Fisher,  G.  E.,  266. 

Flamsteed,  242. 

Florido.    See  Floridus. 

Floridus,  224,  225. 

Foot,  175,  176. 

Fourier,  240. 

Fractions,  20-24,  38-41,  82,  98,  106, 
112,  114,  122,  150-155,  181-183,  185, 
201,  202,  207.  See  Decimal  frac- 
tions. 

Franpais,  243. 


316 


INDEX 


Friedlein,  8,  14,  26,  32,  37-40,  64,  116, 

117,  147. 

Frisius.    See  Gemma-Frisius. 
Froebel,  v. 
Frontinus,  91,  92. 
Furlong,  177. 

Galileo,  155,  250. 

Galley  method  of  division,  148,  149. 
See  Scratch  method. 

Gallon,  172,  176,  177. 

Garnett,  16. 

Gauss,  31,  36,  74,  244,  247,  273,  274, 
279,  283. 

Geber,  107.    See  Dschabir  ibn  Aflah. 

Gellibrand,  163. 

Geminus,  52,  84. 

Gemma-Frisius,  203. 

Geometry,  in  Egypt,  44-46 ;  in  Baby- 
lonia, 43;  in  Greece,  30,  46-89;  in 
Rome,  89-92 ;  in  India,  93,  122-124 ; 
in  Arabia,  125-129;  Middle  Ages, 
131-138;  in  England,  135,  205,206, 
208,  245,  246,  259,  260,  263,  267,  281- 
289;  Modern  Times,  230,  245-289; 
modern  synthetic,  252-259 ;  editions 
of  Euclid,  245-247,  266,  282,  288; 
modern  geometry  of  triangle,  259- 
263 ;  non-Euclidean  geometry,  266- 
275 ;  text-books,  275-289. 

Gerard  of  Cremona,  118,  133, 134. 

Gerbert,  106,  114-119,  131,  132, 147. 

Gergonne,  253,  258,  260. 

Gerhardt,  118,  167,  194. 

Gibbs,  J.  W.,  245. 

Girard,  A.,  153,  230,  231,  233,  234,  247. 

Glaisher,  163. 

Golden  rule,  185,  189,  193,  201.  See 
Rule  of  Three. 

Golden  section,  63. 

Gow,  19,  20,  26,  29,  31-33,  36,  44,  47, 
50,  55-57,  60-62,  65,  67,  76,  78,  81,  85, 
133, 136. 

Graap,  50. 

Grain  of  barley,  170 ;  of  wheat,  172, 
173, 174. 

Grammateus,  141. 

Gramme,  171. 

Grassmann,  245. 

Graves,  55,  241. 

Grebe  point,  262. 


Greeks,  1,  4,  7,  8,  10,  20,  26-37,  46-89, 
91, 94, 102, 123, 126, 171, 175, 248,  263. 

Greenwood,  I.,  217. 

Greenwood,  S.,  217. 

Gregory,  D.,  245. 

Gregory,  D.  F.,  244. 

Gregory,  James,  242. 

Groat,  178 ;  derivation  of  word,  170. 

Gromatici,  89-92. 

Grube,  212. 

Grynaeus,  245. 

Gubar-numerals,  13,  14,  16,  17,  110. 

Guinea,  derivation  of  word,  170. 

Gunter,  163, 179,  199. 

Giinther,  8,  13,  17,  117,  132,  133,  135, 
137,  159,  194,  220,  221,  247,  248,  250, 


Hachette,  257. 

Haddschadsch  ibn  Jusuf  ibn  Matar, 
126. 

Halifax,  John,  122. 

Hallam,  228. 

Halliwell,  122,  133,  135,  181. 

Halsted,  67,  68,  70,  71,  74, 77, 246,  266, 
268,  270-275. 

Hamilton,  W.  R.,  55, 245.    W.,  304. 

Hankel,  5,  26,  36,  40,  47,  50, 51,  52,  56, 
59,  60-62,  70,  72,  89,  90,  96,  97,  100, 
103,  105-107,  116,  121,  129,  132,  133, 
137, 143,  230,  240,  244,  287,  288. 

Hardy,  A.  S.,  243. 

Harpedonaptae,  45. 

Harriot,  157,  231,  232,  234. 

Harrow,  204,  207. 

Harun  ar-Raschid,  126. 

Hassler,  215. 

Hastings,  205. 

Hatton,  156, 195, 199,  202. 

Hawkins,  190,  193. 

Hay  ward,  R.  B.,  241. 

Heath,  33-36. 

Hebrews,  44,  49, 175. 

Heckenberg,  144. 

Hegesippus,  222. 

Heiberg,  30,  67,  69,  75,  84. 

Heiberg  and  Menge,  70,  71,  246. 

Helicon,  63. 

Helmholtz,  275. 

Hendricks,  227. 

Heurici  and  Treutleiu,  280. 


INDEX 


317 


Henry  I.,  175, 186. 

Henry  VII.,  167. 

Henry  VIII.,  169, 172,  246. 

Heppel,  v,  206. 

Hermite,  227. 

Hermotimus,  63. 

Herodianic  signs,  7. 

Herodianus,  7. 

Herodotus,  27. 

Heron  of  Alexandria,  79-82,  90,  102, 

113,  127,  250. 
Heron    the    Elder.      See    Heron    of 

Alexandria. 
Heron  the  Younger,  80. 
Heronic    formula,   80,    90,   123,  128, 

134. 

Heteromecic  numbers,  29. 
Hexagramma  mysticum,  255. 
Hieratic  symbols,  7. 
Hieroglyphics,  6,  45. 
Hill,  J.,  199,  201,  203. 
Hill,  T.,  219. 
Hindu  notation,  11,  13-18,  145,  154, 

155. 
Hindu  numerals,  12-18,  107, 119,  121, 

179,  180,  181,  187. 
Hindu  proof,  106.    See  Casting  out 

the  9's. 
Hindus,  8,  9,  11, 12,  15,  24,  82,  88,  93- 

103,  113,  116,  119,  122-124,  128,  136, 

143,  145,  147,  149,  150,  170,  215,  221, 

227,  233,  238,  250. 
Hipparchus,  79,  84. 
Hippasus,  51,  52. 
Hippias,  55,  56. 
Hippocrates  of  Chios,  56,  57,  59,  60, 

61,65. 

Hirst,  279,  284,  286. 
Hodder,  191,  217. 
Hoefer,  133. 
Hoernle,  94. 
Hoffmann,  50,  128. 
Hoffmann's  Zeitschrift,  280. 
Holywood,  122. 
Homological  figures,  253. 
Homology,  law  of,  288-. 
Hook,  208. 
Horace,  40. 
Homer,  226,  240. 
Homer's    method,    240,    241.      See 

Horner. 


Hoiiel,  278. 

Hughes,  T.,  232. 

Hultsch,  65,  81. 

Hume,  167,  168. 

Hunain,  127. 

Hunain  ibn  Ishak,  127. 

Hutt,  264. 

Button,  189,  243,  244,  264. 

Huxley,  283,  306. 

Hypatia,  87. 

Hyperbolic  logarithms,  166, 167.    See 

Natural  logarithms. 
Hypothetical    constructions,  73,    74, 

277. 
Hypsicles,  28,  31,  67,  79, 127. 

lamblichus,  27,  30,  33,  82, 108. 
Ibn  Albanna,  106,  111.  150. 
Imaginary  geometry,  272.    See  Non- 
Euclidean  geometry. 
Imaginary    quantities,   227,  231-233, 

236,  243-245. 

Inch  (uncia),  40,  171, 175. 
Incommensurables,  63,  66,   72,    266, 

278,  280,  285-288.     See  Irrational 

quantities. 
India.    See  Hindus. 
Indices.     See  Exponents. 
Infinite,  59,  70,   136,  250,  252,    278; 

symbol  for,  237. 
Infinite  series.    See  Series. 
Infinitesimal,  136. 
Insurance,  200. 

Interest,  100,  121, 151, 167,  168, 198. 
Inversion,  99. 

Involution  of  points,  87,  252,  253. 
Ionic  School,  47^9. 
Irrational  quantities,  29,  51,  67,  72, 

76, 101,  102,  103,  108,  287.    See  In- 

commensurables. 
Isidorus,  88. 

Isidorus  of  Carthage,  92,  111. 
Isoperimetrical  figures,  79,  136. 
Italian  method  of  division,  213. 

Jakobi,  C.  F.  A.,  260,  261. 
Joachim,  251.     See  Rhaeticus. 
John  of  Palermo,  120. 
John  of  Seville,  118, 119, 150. 
Johnson  -,  3. 
Jones,  283. 


318 


INDEX 


Jones,  W.,  200. 

Jonquieres,  de,  265. 

Jordanus  Nemorarius,  134,  135,  142. 

Joseph  Sapiens,  117. 

Joseph's  Play,  222. 

Josephus,  222. 

Kambly,  280. 

Kant,  277, 

Kastner,  127,  181,  207,  279. 

Kelly,  168,  169,  170,  174, 196. 

Kempe,  264. 

Kepler,  136,  155,  166,  167,  237,  247, 

249,  252,  269. 
Kersey,  153,  194,   196,  199,  201,  203, 

220,  221. 

Kettensatz,  196.    See  Chain-rule. 
Kiugsley,  87. 
Kircher,  247. 

Klein,  70,  74,  249,  264,  265. 
Koppe,  K.,  280. 
Krause,  247. 
Kronecker,  227. 
Kruse,  280. 


Lacroix,  S.  F.,  268,  276. 

Laertius,  Diogenes,  28,  47,  61. 

Lagny,  de,  241,  242. 

Lagrange,  36,  214,  240,  270,  283. 

Lahire,  de,  252,  253. 

Lambert,  242,  248,  269,  274. 

Lange,  J.,  259. 

Langley,  E.  M.,  213,  241. 

Langley  and  Phillips,  288. 

Laplace,  155,  269. 

La  Roche,  142,  144. 

Larousse,  276. 

Lefere,  142. 

Legendre,  31,  243,  269,  270,  271,  272, 

276,  277,  278,  283,  286. 
Leibniz,  195,  254,  262. 
Lemma  of  Menelaus,  83. 
Lemoine,  E.,  259,  262. 
Lemoine  circles,  262. 
Lemoine  point,  262. 
Leodamas,  63. 
Leon,  63,  65. 

Leonardo  da  Vinci,  141,  248,  249. 
Leonardo  of  Pisa,  24,  76,  119-121, 134, 

135,  139,  141,  150,  196,  222. 


Leudesdorf ,  248. 

Leyburn,  203. 

Limits,  77,  78,  250,  278,  287. 

Lindemann,  243. 

Linkages,  263,  264. 

Lionardo  da  Vinci.    See  Vinci. 

Lippich,  265. 

Littre,  E.,  144. 

Lobatchewsky,  86,  127,  266,  272-275, 

279. 
Local  value,  principle  of,  8,  9,  10,  11, 

12,  15,  42,  95, 105,  118. 
Locke,  144. 

Logarithmic  series,  167. 
Logarithmic  tables,  160-166. 
Logarithms,  155-167, 187,  199,  229. 
Lorenz,  269. 
Loria,  25,  47,  48,  50,  56,  59,  65,  67,  76, 

277-280,  283. 
Lowe,  189. 
Liibsen,  279. 

Luca  Paciuolo.    See  Pacioli. 
Lucas  di  Burgo.    See  Pacioli. 
Lucian,  31. 

Ludolph  van  Ceulen,  242. 
Ludolph's  number,  242. 
Ludus  Joseph,  222. 
Lune,  57. 
Lyte,  152. 


Macdonald,  156, 160,  194,  206. 

Machin,  242. 

Mackay,  259,  262,  288. 

Mackay  circles,  263. 

Maclaurin,  208,  244,  256. 

Magic  squares,  221. 

Mahavira,  95. 

Malcolm,  200. 

Malfatti,  226. 

Marcellus,  75,  78. 

Marie,  80,  133,  199,  252,  264 

Marsh,  200. 

Martin,  B.,  194, 199. 

Martin,  H.,  218. 

Mascheroni,  264. 

Mathematical    recreations,    24,    113i 

219-223. 

Matthiessen,  23,  227,  234. 
Maudith,  137. 
Maxwell,  77. 


INDEX 


319 


McLellan  and  Dewey,  212. 

Mean  proportionals,  56. 

Measurer  by  the  ell,  176. 

Medians  of  a  triangle,  134. 

Meister,  247. 

Mellis,  188. 

Mensechmus,  63. 

Menelaus,  82,  85,  129,  133 ;  lemma  of, 
83. 

Menge.    See  Heiberg  and  Menge. 

Mental  arithmetic,  204,  219. 

Mercator,  N.,  167. 

Merchant  Taylors'  School,  204,  206, 
207. 

Methods.  See  Teaching  of  Mathe- 
matics. 

Metric  System,  152,  171,  178,  179,  215. 

Me'ziriac,  220,  222. 

Million,  143,  144. 

Minus,  symbol  for,  141, 184,  228.  See 
Subtraction. 

Minutes,  10. 

Mobius,  247,  258. 

Moivre,  de,  208. 

Moneyer's  pound,  168.  See  Tower 
pound. 

Monge,  257. 

Morgan,  de.     See  De  Morgan. 

Moschopulus,  221. 

Motion  of  translation,  70,  135,  157; 
of  reversion,  70;  circular  and  recti- 
linear, 263,  264. 

Muhammed  ibn  Musa  Alchwarizml, 
104-109,  118,  128,  129,  133,  206. 

Miiller,  F.,  107. 

Miiller,  H.,  268,  280. 

Miiller,  John,  250.  See  Regiomon- 
tanus. 

Multiplication,  26,  96-98,  101,  105, 
110,  115,  140,  145-147,  156,  189,  194, 
195,  213,  214,  234 ;  of  fractions,  21, 
22,  181,  182. 

Multiplication  table,  32,  39,  40,  115, 
117,  147,  148,  181. 

Murray,  173. 

Musa  ibn  Schakir,  128. 

Mystic  hexagon,  255. 


Naperian  logarithms,  159. 
Napier,  M.,  154. 


Napier,  J.,  149, 153,  155-167,  194,  206, 

229,  251. 

Napoleon  I.,  69,  257,  264. 
Nasir  Eddin,  127,  128,  130,  131,  270. 
Natural  logarithms,  159, 160, 164, 166. 
Naucrates,  65. 
Negative  quantities,  35,  88,  101, 102, 

107,  227,  230-233,  243,  244,  258. 
Nemorarius,  134,  135,  142.    See  Jor- 

danus  Nemorarius. 
Neocleides,  63. 
Neper.    See  Napier. 
Nesselmann,  30,  31,  37,  72,  108,  109. 
Neuberg  circles,  263. 
Newcomb,  68,  288. 
Newton,  55,  77,  167,  170,  194,  208,  209, 

226,  237-240,  243,  255,  273. 
Nicolo  Fontana.    See  Tartaglia. 
Nicomachus,  26,  31-33,  41,  91,  92. 
Niedermiiller,  214. 
Nine-point  circle,  259,  260. 
Nixon,  285,  286,  288. 
Non-Euclidean  geometry,  266-275. 
Norfolk,  180,  181. 
Norton,  R.,  152. 
Notation,  of  numbers  (see  Numerals, 

Local  value) ;  of  fractions,  21, 26, 98, 

106,  181 ;  of  decimal  fractions,  152- 

154 ;  in  arithmetic  and  algebra,  23, 

30,  35,  36,  98,  101,  108-111,  122,  184, 
194, 195,  228,  229,  234,  235,  237,  239. 

Number,  concept  of,  29,  35,  232,  233. 
See  Negative  quantities,  Imagina- 
ries,  Irrationals,  Incommensu- 
rables. 

Numbers,  amicable,  29;  cubical,  32; 
defective,  29,  113;  excessive,  29, 
111;  heteromecic, 29 ;  prime,  30,  31, 
74;  perfect,  29,  111,  113;  polygonal, 

31,  32 ;  triangular,  29. 
Number-systems,  1-18. 
Numerals,  6,  7,  8,  12-16,  32,  38,  107, 

119, 121,  180. 
Numeration,  1-5,  8,  12-18, 142-145. 


Observation  in  mathematics,  74,  77, 

78,  151,  155,  283,  288. 
Octonary  numeration,  3. 
CEnopides,  46. 
Oinopides.    See  CEnopides. 


320 


INDEX 


Ohm,  M.,  244. 

Oldenburg,  237. 

'Omar  Alchaijami,  110. 

Oresme,  122,  234. 

Otho,  V.,  251. 

Ottaiano,  87. 

Oughtred,  149,  153,  154, 187, 194, 195, 

203,  234. 

Ounce  (uncia),  40,  112,  171,  172, 173. 
Oxford,  university  of,  137,  138. 

IT,  44,  49,  73,  76, 123, 128, 145, 241-243. 

See  Squaring  of  the  circle. 
Pacciuolus.    See  Pacioli. 
Pacioli,  105, 139, 141-143, 145, 146-148, 

181,  182,  188, 196,  208,  214,  222,  232, 
238,  245. 

Paciuolo.    See  Pacioli. 

Padmanabha,  95. 

Palatine  anthology,  33,  34. 

Palgrave,  186,  256. 

Palms,  175. 

Paolis,  R.  de,  279. 

Paper  folding,  265. 

Pappus,  65,  79,  82,  86-89,  248,  255. 

Parallel  lines,  43,  52,  85,  86,  127,  252, 

266-275,  280,  282. 
Parallel-postulate,  70,  71,  85,  86,  127, 

266-275,  279,  282,  283,  289 ;  stated, 

71. 

Parley,  Peter,  218. 
Pascal,  238,  252-255. 
Pascal's  theorem,  255,  257. 
Pathway  of  Knowledge,  173, 174, 188, 

189. 

Peacham,  205,  210. 
Peacock,  2, 121, 143,  145-150, 153, 181, 

182,  186,  187,  193,  198,  220,  244. 
Peaucellier,  263, 264. 
Pedagogics.    See  Teaching. 
Peirce,  B.,  245. 

Peirce,  C.  S.(  67,  70. 

Peletier,  230,  266. 

Pentagram-star,  136. 

Pepys,  192. 

Perch,  178. 

Perfect  numbers,  29,  111,  113. 

Perier,  Madam,  254. 

Perry,  J.,  291. 

Pestalozzi,  v,  197,  203,  211,  212, 219. 

Petrus,  136. 


Peuerbach.  See  Purbach. 

Peurbach.  See  Purbach. 

Peyrard,  71. 

Philippus  of  Mende,  63. 

Philolaus,  29,  53. 

Phoenicians,  208. 

Picton,  180. 

Pike,  214,  216-218. 

Pitiscus,  203,  251. 

Planudes,  16,  95,  213. 

Plato,  26,  29,  46,  50,  53,  54,  56,  60-63, 
65, 114,  208. 

Plato  of  Tivoli,  118,  124,  130,  134. 

Plato  Tiburtinus.   See  Plato  of  Tivoli. 

Platonic  figures,  65,  73,  78. 

Platonic  school,  53,  60-63. 

Playfair,  270,  271,  282. 

Plus,  symbol  for,  141,  184,  228.  See 
Addition. 

Plutarch,  47. 

Poinsot,  247. 

Polars,  257,  258. 

Polygonal  numbers,  31,  32. 

Poncelet,  248,  252,  253,  257,  258, 
260. 

Pons  asinorum,  137. 

Porisms,  74. 

Porphyrius,  82. 

Position,  principle  of.  See  Local 
value. 

Postulates,  46,  54,  61,  67,  69,  70,  71, 
73,  89,  134,  266-275,  279,  281. 

Pothenot,  251. 

Pothenot's  problem,  251. 

Pott,  4. 

Pound,  40, 168-175,  178,  185,  218. 

Practice,  22,  189,  196. 

Primes,  30,  31,  74. 

"  Problems  for  Quickening  the  Mind," 
113,  131,  220. 

Proclus,  47,  61,  63,  70,  74,  85,  88, 
247. 

Progressions,  arithmetical,  9,  23,  24, 
31,  100,  158, 160, 181,  185,  237 ;  geo- 
metrical, 9,  23,  24, 100, 158, 160, 181, 
237. 

Projection,  258. 

Proportion,  30,  31,  47,  53,  57,  65,  67, 
72,  73,  110,  196,  197,  202,  234,  277, 
282,  285,  286.  See  Rule  of  Three. 

Ptolemseus.     See  Ptolemy. 


INDEX 


321 


Ptolemy,  28,  82,  84,  104, 123, 124, 126, 
129,  130,  270. 

Ptolemy  L,  64. 

Public  schools  in  England,  204. 

Purbach,  140,  148,  149,  213,  250. 

Pythagoras,  46,  48-53, 123,  136 ;  theo- 
rem of, 45, 49-51, 56, 66, 123, 128, 137. 

Pythagorean  school,  49-53. 

Pythagoreans,  14,  27,  29,  30, 49-53, 66, 
136,  250. 


Quadratic  equations.    See  Equations. 
Quadratrix,  55. 

Quadrature  of  the  circle.     See  Squar- 
ing of  circle. 
Quadratures,  167,  237. 
Quadrivium.    See  Quadruvium. 
Quadruvium,  92,  111,  114. 
Quick,  v,  212. 
Quinary  scale,  1,  3,  4. 
Quotient,  37, 115, 183. 

Radian,  251. 

Radical  sign,  228,  229,  235,  236. 

Raleigh,  W.,  231. 

Ralphson,  55. 

Ramus,  247,  275. 

Ratio,  32,  51,  63,  69,  76,  87, 167,  188, 

194,  195,  234. 
Raumer,  286. 
Reciprocal  polars,  258. 
Recorde,  118.  147,  149,  183-188,  194, 

196, 198,  206,  208,  228,  234,  281. 
Reddall,  204,  207. 
Reductio  ad  absurdum,  58,  60,  62. 
Rees,  A.,  166. 
Rees,  K.  F.,  196,  197. 
Reesischer  Satz,  196.    See  Chain-rule. 
Regiomontanus,  108,  136, 140, 181, 208, 

229,  247,  250,  251. 
Regular  solids,  52,  62,  65,  67,  73,  78, 

128,  136,  249. 
Reiff ,  238. 
Rhseticus,  187,  251. 
Rheticus.    See  Rhaeticus. 
Rhetorical  algebras,  108,  120. 
Rhind  papyrus,  19.    See  Ahmes. 
Richards,  E.  L.,  74,  268. 
Riemann,  31,  275,  283. 
Riese,  Adam,  31,  142, 192,  196,  2ia 


Rigidity  of  figures,  280. 

Rigidity  postulate,  71. 

Roberval,  209. 

Robinson,  216,  267. 

Roche.    See  La  Roche. 

Rodet,  94. 

Rods,  Napier's,  156, 157. 

Rohault,  65. 

Roman  notation,  4,  8,  11,  1?1,  180. 

Romans,  1,  4,  8,  20,  37-42,  82,  88-92 

116,  122,  131,  135,  168,  175. 
Romanus,  A.,  242,  251. 
Roods,  178. 
Root  (see  Square  root,  Cube  root), 

36,  72,  76,  101,  102,  106,  108,  110, 

119, 150,  151,  200,  227-233,  235,  236, 

241. 

Rosenkranz,  208. 
Rouche  and  C.  de  Comberousse,  278, 

287. 

Row,  S..  265. 

Rudolff,  141,  143,  151,  228,  229,  232. 
Ruffini,  226. 
Rugby,  204,  207. 
Rule  of  Falsehode,  184.     See  False 

position. 

Rule  of  four  quantities,  129. 
Rule  of  six  quantities,  83, 129. 
Rule  of  Three,  100, 106,  121, 185,  193- 

196, 198,  201,  202.    See  Proportion. 
Ruler  and  compasses,  54, 135, 248, 249, 

264. 

Ruler,  graduated,  134. 
Rutherford,  242. 


Saccheri,  267,  269,  271,  274,  278. 
Sacrobosco,  122. 
Sadowski,  148,  213,  215. 
Salvianus  Julianus,  41. 
Sand-counter,  29. 
Sannia  and  E.  D'Ovidio,  279,  287. 
Sanscrit  letters,  16. 
Saunderson,  244. 
Savile,  281,  282. 
Schilke,  79. 
Schlegel,  280. 

Schlussrechnung,  196, 197.    See  Anal- 
ysis in  arithmetic. 
Schmid,  K.A.,  143,  286. 
Schmidt,  F.,  273. 


322 


INDEX 


Schotten,  H.,  230. 

Schreiber,  H.,  141. 

Schroter,  265. 

Schubert,  H.,  249. 

Schulze,  166. 

Schwatt,  266. 

Scratch  method  of  division,  148, 149, 

192,  215 ;  of  multiplication,  105. 
Scratch  method  of  square  root,  150. 
Screw,  55. 

Secant,  130,  137,  251. 
Seconds,  10. 
Seneca,  65. 
Serenus,  82-84. 
Series,  32,  110,  120,  167,  229,  238,  239, 

242,  243.    See  Progression. 
Serret,  227. 
Servois,  248,  257. 
Sexagesimal  fractions,  10,  20, 106, 119, 

120,  151. 
Sexagesimal  scale,  6,  9,  10,  28,  43,  79, 

84,  85,  124,  163. 
Sextus  Julius  Africanus,  86. 
Shanks,  145,  242. 
Sharp,  A.,  241,  242. 
Sharpless,  204,  207. 
Shelley,  173,  187, 199. 
Shillings,  168,  169,  170,  185,  218. 
Sieve  of  Eratosthenes,  31. 
Simplicius,  58. 

Simpson,  T.,  243,  244,  267,  284. 
Simson,  R.,  67,  69,  71,  74,  246,  256, 

282,  285. 
Sine,  85,  124,  129,  130,  159,  162, 163, 

251 ;  origin  of  word,  124,  130. 
Singhalesian  signs,  12. 
Single  position,  196.    See  False  posi- 
tion. 

Slack,  Mrs.,  192,  217. 
Sluggard's  rule,  147. 
Snellius,  251. 
Socrates,  60,  65. 
Sohncke,  83. 

Solids,  regular.    See  Regular  solids. 
Solon,  7,  28. 
Sophists,  26,  53-60,  276. 
Sosigenes,  91. 
Speidell,  E.,  165. 
Speidell,  J.,  164,  165. 
Spencer,  v. 
Spenser,  172. 


Sphere,  52,  62,  63,  78,  79,  249. 
Spherical  geometry,  73,  83,  125,  156 , 

247.    See  Sphere. 
Square  root,  28,  76,  101,  106,  110,  150, 

151,  154,  185. 
Squaring  of  the  circle,  54,  55,  57,  58. 

200,  226,  249,  271,  282.     See  IT. 
Stackel.    See  Engel  and  Stackel. 
Star-polyaedra,  247,  250. 
Star-polygons,  136,  247. 
Staudt,  von,  258,  259,  265. 
Steiner,  248,  258,  264. 
Stephen,  L.,  133. 

Sterling,  172;  origin  of  word,  170. 
Stevin,  122,  149,  151-153, 157,  234,  235. 
Stewart,  M.,  256. 
Stifel,  141,  143,  149,  157,  158,  208,  229, 

232,  233,  234,  238. 
Stobseus,  65. 

Stone,  201,  240,  262,  267,  269. 
Sturm,  253. 
St.  Vincent,  167. 
Subtraction,  26,  37,  38,  96,  101,  105, 

110,  115,  145. 
Surds,  150. 
Surya-siddhanta,  12. 
Surveying,  89,  91,  136,  178. 
Suter,  128-130,  133,  137. 
Sylvester,  22,  264,  283,  285. 
Symbolic  algebras,  109. 
Symmedian  point,  259,  262. 
Syncopated  algebras,  108. 
Synthesis,  62. 


.Tabit  ibn  Kurrah,  127,  129. 

Tables,  logarithmic,  160-166;  multi- 
plication, 32,  39,  40,  115,  117,  147, 
148,  181 ;  trigonometric,  79,  84,  124, 
125,  130,  159,  250,  251. 

Tacquet,  66,  188. 

Tagert,  70. 

Tangent  in  trigonometry,  130,  137, 
250,  251. 

Tannery,  47,  56,  81,  128. 

Tartaglia,  139,  145,  150,  188,  193,  196, 
198,  208,  220,  222,  224,  225,  228,  248. 

Taylor,  B.,  208. 

Taylor,  C.,  252,  257,  264. 

Taylor,  H.  M.,  68,  288. 

Taylor  circles,  263. 


INDEX 


323 


Teaching,  hints  on  methods  of,  v, 
17,  25,  41,  74,  78,  114,  138,  151,  155, 
183,  185,  189,  197,  198,  211,  212,  233, 
236,  238,  239,  277,  281,  288. 

Telescope,  155,  232. 

Thales,  46-49,  52. 

Theaetetus,  63,  64,  66. 

Theodorus,  60. 

Theodosius,  133. 

Theon  of  Alexandria,  28,  69,  79,  82, 
87,  91. 

Theon  of  Smyrna,  33,  82. 

Theory  of  numbers,  37,  51,  82,  112. 

Theudius,  63,  65. 

Thompson,  T.  P.,  279. 

Thymaridas,  33,  108. 

Timaeus  of  Locri,  60. 

Timhs,  204,  205,  207,  210. 

Todhunter,  62,  68,  69. 

Ton  stall,  16,  143,  149,  180-183,  185, 
187,  188,  203,  206,  208. 

Torporley,  251. 

Tower  pound,  168,  171,  172,  174. 

Transversals,  83,  252,  255,  258. 

Treutlein,  13. 

Triangular  numbers,  29. 

Trigonometry,  79,  81,  84,  85,  123-125, 
129-131,  136,  139,  155,  156,  159,  165, 
230,  243,  250,  251,  270. 

Trisection  of  an  angle,  34,  55,  134, 
135,  226. 

Trivium,  92,  114. 

Troy  pound,  169,  170,  172,  174 ;  deri- 
vation of  word,  172. 

Tucker,  262. 

Tucker  circle,  262,  263. 

Tycho  Brahe,  250. 

Tylor,  3,  4,  17,  204. 

Tyndale,  183. 


Unger,  16,  121,  140,  142,  144,  147,  148, 
181,  191,  194,  195,  204,  207,  212,  215. 
University  of  Oxford,  137,  138. 
University  of  Paris,  137. 
University  of  Prague,  137. 
Usury,  168. 


Vacquant,  Ch.,  278. 
Valentin,  87. 


Van  Ceulen,  242. 

Varignon,  267. 

Varro,  91. 

Vasiliev,  272. 

Venturi,  81. 

Veronese,  G.,  279. 

Versed  sine,  124. 

Vesa,  242. 

Victorius,  40. 

Vieta,  79,  109,  187,  226,  228-232,  233 

234,  238,  240,  242,  250,  251. 
Vigarie",  E.,  259. 
Vigesimal  scale,  2,  3,  4. 
Villefranche,  142. 
Vinci,  Leonardo  da,  141,  248,  249. 
Vlacq,  163,  167. 
Von  Staudt,  258,  259,  265. 


Wafa.    See  Abu'l  Waf&. 

Wagner,  U.,  140. 

Walker,  F.  A.,  169. 

Walker,  J.  J.,  157. 

Wallace  line  and  point,  259. 

Wallis,  123,  128,  149,  153,   164,  200, 

208,  209,  236,  237,  269,  274. 
Wantzel,  227. 
Ward,  200,  209. 
Warner,  W.,  232. 
Webster,  W.,  210. 
Weights  and  measures,  9,  140,  152, 

166-179 ;  in  United  States,  215-217. 
Weissenborn,  76,  106,  117,  255. 
Wells,  205. 
Wessel,  C.,  243. 
Whewell,  281. 
Whiston,  66,  239. 
Whitehead,  246. 
Whitley,  260. 
Whitney,  12. 
Widmann,  140, 141,  196. 
Wiener,  H.,  265. 
Wilder,  194. 
Wildermuth,  143,  153,  154,  189,  195, 

222. 

William  the  Conqueror,  168, 171. 
Williamson,  246. 
Wilson,  J.  M.,  268,  283. 
Winchester  bushel,  176.    See  Bushel 
Winchester  (school) ,  204,  206. 
Windmill,  137. 


324 


INDEX 


Wingate,  153,  173,  187,  194,  195,  196, 

199,  201,  202,  203,  220,  221. 
Wipper,  50, 128. 
Wolf,  194,  266. 
Wolfram,  166. 
Wopcke,  14,  15. 
Wordsworth,  C.,  244. 
Worpitzky,  280. 
Wright,  153,  200,  206. 


Xenocrates,  61. 


Yancos,  145. 
Yard,  175,  176. 
Young,  6. 


Zambertus,  245. 

Zeno,  58,  59. 

Zenodorus,  79,  136,  247. 

Zero,  10,  11,  12,  14,  15,  42,  96,  99,  115 

118,  119,  143, 153. 
Ziwet,  280. 
Zschokke,  215. 


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